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Transition of interactions between a cuckoo and its host: Fluctuating between parasitism and mutualism
Applied Mathematics and Computation 273 (2016) 664–677
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Transition of interactions between a cuckoo and its host: Fluctuating between parasitism and mutualism Shikun Wang a, Yuanshi Wang b,∗ a b
School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, PR China School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, PR China
a r t i c l e
i n f o
Keywords: Cooperation Persistence Coexistence Stability Bifurcation
a b s t r a c t This paper considers crow–cuckoo–cat systems in which crows are the host, cuckoos are the parasite, cats are the predator of crow nests, and cuckoo chicks in the nests reveal a mix of caustic and repulsive compounds that repel and deter cats. An over 16-year observation on the crow–cuckoo–cat system shows that the parasite and its host can be mutualistic in the presence of predators (Canestrari et al., 2014). In order to exhibit mechanisms by which the mutualism occurs, a crow–cuckoo–cat model is formed in this work. Global dynamics of the model demonstrate that cats’ predation ability and converting efficiency are crucial to the occurrence of mutualism. When the predation/efficiency is low, cats would be driven into extinction by cuckoos. When the predation/efficiency is intermediate, the three species coexist and interaction outcomes between crows and cuckoos change from parasitism to commensalism and mutualism as the deterrence effect increases. When the predation/efficiency is high, cats would drive cuckoos into extinction if the deterrence is weak. If the deterrence is strong, either cuckoos or cats go extinct, where the species with higher initial density persists. Therefore, results in this work show the empirical observation by Canestrari et al. (2014), and predict situations that have not been observed by Canestrari et al. (2014). Numerical simulations display that the crow–cuckoo–cat model fits the observation well. © 2015 Elsevier Inc. All rights reserved.
1. Introduction An over 16-year observation on carrion crows (Corvus corone corone) and the great spotted cuckoo (Clamator glandarius) shows that a parasite (cuckoo) can provide a benefit to its host (crow) by deterring crow nest predators [4]. The cuckoo is called a parasite of the crow since it lays eggs in the nest of the crow, which breeds the unrelated chicks and suffers the loss of its brood. Canestrari et al. [4] studied crow nests in northern Spain and found that the nests parasitized by cuckoos are more successful in crows’ raising their own chicks than non-parasitized nests. This is due to predator repellence by a malodorous cloacal secretion that parasitic chicks release. The cuckoo secretion is a mixture of caustic and repulsive compounds, which are known to repel crow nest predators such as corvids, raptor birds and quasi-feral cats. A cat is called quasi-feral if it is free-ranging and hunts year-round but could be attracted with food. For convenience, we use cats to represent the predators in this work. The long-term observation by Canestrari et al. [4] displays that although cuckoos decreased host reproductive success during seasons with low nest predation, parasitized nests produced more fledglings than non-parasitized nests during seasons with
∗
Corresponding author. Tel.: +86 20 8403 4848; fax: +86 20 8403 7978. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.amc.2015.10.067 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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high nest predation. Thus, interaction outcomes between the crow and cuckoo fluctuate between parasitism and mutualism each season, depending on the intensity of predation pressure. While the novel mutualism between the parasite and its host is observed, mechanisms by which the mutualism can occur need to be demonstrated. Parasitism/predation systems of two species have been studied by a series of significant works. Volterra [18] established a predator–prey model to describe interactions between the predator fish and prey fish in Adriatic during the First World War. Functional responses in the model are linear and all positive solutions (except the equilibrium) are shown to be periodic. Since the time averages of all periodic solutions correspond to components of the unique positive equilibrium, dynamics of the model well characterize the mechanism by which the amount of predator fish during the First World War was considerably higher than those in the years before. The model is also called the Lotka–Volterra predator–prey model because of the important contribution of Lotka [14]. Moreover, the model is extended to describe the predation system where there exists intraspecific competition in each species. The extended model is structurally stable and all of its positive solutions converge to an equilibrium (see for example, [10]). The two-species model is expanded to describe predation systems with multiple species. Freedman and Waltman [7] considered a general predator–prey system of three species, in which two predators have a common prey. A general mathematical criterion is given for persistence of the three-species system. Cantrell et al. [5] studied a specific two-consumer and one-resource system in which there is no interaction between the consumers. Theoretical analysis and numerical simulations showed that the two consumers can coexist upon the single resource at steady states or in periodic oscillations as the environmental condition varies. Lisena [13] considered a two-predator and one-prey system with periodic coefficients, in which there is competition between the predators. Suitable average conditions are given to extinction of the competitively inferior species. For more relevant works, see Armstrong and McGehee [1], Smith and Thieme [15], Hsu et al. [11], Wang and Wu [16], Wang et al. [17], etc. While these works focus on coexistence of consumers, there is little work on parasitism-predation models in which the parasite and its host can be mutualistic in the presence of predators. Thus, modelling the crow–cuckoo–cat system and exhibiting mechanisms by which the mutualism between the crow and cuckoo can occur is necessary. Motivated by the empirical observation and Lotka–Volterra predator–prey model, we expand the model to describe the parasitism-predation system consisting of crows, cuckoos and cats. In the system, crows are the host, cuckoos are the parasites, cats are the predators of crows, and cuckoo chicks reveal a mix of caustic and repulsive compounds that repel and deter cats. Global dynamics of the crow–cuckoo–cat model demonstrates that cats’ predation ability and converting efficiency are crucial to the occurrence of the mutualism between crows and cuckoos. When the predation/efficiency is low, cats would be driven into extinction by cuckoos. When the predation/efficiency is intermediate, the three species coexist and interaction outcomes between crows and cuckoos change from parasitism to commensalism and mutualism as the deterrence effect increases. When the predation/efficiency is high, cats would drive cuckoos into extinction if the deterrence is weak. If the deterrence is strong, either cuckoos or cats go extinct, where the species with higher initial density persists. Therefore, results in this work show the empirical observation by Canestrari et al. [4], and predict situations that have not been observed by Canestrari et al. [4]. Simulations illustrate the results. The paper is organized as follows. The crow–cuckoo–cat model is formed in Section 2. In Section 3, we consider stability of equilibria and in Section 4, we show persistence of the system. We exhibit occurrence of mutualism in Section 5, and a discussion of the results is in Section 6. 2. The crow–cuckoo–cat model In this section, a three-species model is formed to characterize the crow–cuckoo–cat system. Since cuckoo chicks consume food provided by adult crows, the crow-cuckoo interaction is parasitism, which can be described by a parasitism model. On the other hand, since cats kill and eat crows, the crow–cat interaction is predation, which can be depicted by a predator–prey model. Then the two models are combined to form a crow–cuckoo–cat model, and boundedness of solutions of the three-species model is shown. In the crow–cuckoo interaction, immature cuckoos are bred by crows, which results in a loss of crows in their own brood. Thus the crow-cuckoo interaction is parasitic and we apply the Volterra type model to depict the interaction
u + β1 v du = r1 u 1 − dt K dv = dt
v( − r2 + α1 u − γ1 v)
where variables u and v denote population densities of crows and cuckoos, respectively. Parameter r1 represents the intrinsic growth rate of crows, while K is the carrying capacity. Parameter β 1 denotes the quantity of resources consumed by a cuckoo, and α 1 is the efficiency of the cuckoo in converting the consumption into fitness. Parameter r2 is the mortality rate of cuckoos, while γ 1 represents the intraspecific competition. In the crow–cat interaction, cats are predators and crows are their prey, which can be described by
u du − β2 uw = r1 u 1 − dt K dw = w( − r3 + α2 u − γ2 w) dt
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where variables u and w represent population densities of crows and cats, respectively. Parameter β 2 represents the rate of crows killed by a cat, which denotes predation ability of the cat. Parameter α2 = eβ2 with 0 < e < 1 represents the efficiency of cats in converting the predation into fitness, which is determined by cats’ predation ability (β 2 ) and the converting coefficient (e). Parameter r3 is the mortality rate of cats. In the crow–cuckoo–cat interactions, an immature cuckoo produces a mix of caustic and repulsive compounds, which deters the predation of cats on the crows’ nest. Thus, the cuckoo has a negative effect on the growth of cats. Similar to the Lotka–Volterra predator–prey model, we apply a linear functional response to depict the effect by subtracting a term proportional to v. Moreover, it has been observed that cats do not attack cuckoos with the compounds (page 1351, [4]). By combining the above crow–cuckoo and crow–cat models, we obtain a crow–cuckoo–cat model
u + β1 v du = r1 u 1 − dt K
− β2 uw
dv = v( − r2 + α1 u − γ1 v) dt dw = w( − r3 + α2 u − β3 v − γ2 w) dt For simplicity, we rewrite the crow–cuckoo–cat model by
du = r1 u(1 − c1 u − b1 v − b2 w) dt dv = r2 v( − 1 + a1 u − c2 v) dt dw = r3 w( − 1 + a2 u − b3 v − c3 w) dt
(2.1)
with initial conditions
u(0) ≥ 0,
v(0) ≥ 0, w(0) ≥ 0.
(2.2)
Parameter b3 represents the negative effect of cuckoos on the growth of cats, and all parameters in system (2.1) are positive, which satisfy the following relations
a1 =
α1 r2
, a2 =
α2 r3
, b1 =
β1 K
, b2 =
β2 r1
, b3 =
β3 r3
, c1 =
1 γ1 γ2 , c3 = . , c2 = K r2 r3
We can see that the solutions to the initial value problem (2.1) and (2.2) are nonnegative. We also have the following result on boundedness of the solutions. Proposition 2.1. Solutions of system (2.1) are bounded. Proof. From the first equation of system (2.1), we have
du ≤ r1 u(1 − c1 u) dt so that the comparison principle [6] implies that
lim sup u(t ) ≤ K1 , K1 = t→∞
1 . c1
Thus, for 0 > 0 small, we have u(t ) ≤ 0 + K1 when t is sufficiently large. From the second equation in system (2.1), we have
dv ≤ r2 v(Kˆ2 − c2 v), Kˆ2 = −1 + a1 (0 + K1 ). dt If Kˆ2 ≤ 0, then limt→∞ v(t ) = 0. If Kˆ2 > 0, then
lim sup v(t ) ≤ K2 , K2 = t→∞
Kˆ2 . c2
Thus, we conclude lim supt→∞ v(t ) ≤ |K2 |. Similarly, we obtain
lim sup w(t ) ≤ |K3 |, K3 = t→∞
−1 + a2 (0 + K1 ) . c3
Therefore, solutions of system (2.1) are bounded.
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There are three two-species subsystems to be considered. In this section, we cite some known results on the dynamics of the two-species subsystems. We first consider the parasitism subsystem consisting of crows and cuckoos. Let w = 0. Then system (2.3) becomes the crow-cuckoo system
du = r1 u(1 − c1 u − b1 v) dt dv = r2 v( − 1 + a1 u − c2 v). dt
(2.3)
This is the Lotka–Volterra predator–prey system. There are two equilibria on the axes, namely E0 = (0, 0) and E1 = (1/c1 , 0). E0 is a saddle point with eigenvalues
μ(01) = r1 , μ(02) = −r2 . E1 has eigenvalues
a μ(11) = −r1 , μ(12) = r2 −1 + 1 . c1
(2)
¯ of (2.3) exists if μ1 ¯ v) The positive equilibrium E12 (u,
u¯ =
> 0 with
a1 − c1 b1 + c2 , v¯ = . a1 b1 + c1 c2 a1 b1 + c1 c2
(2.4)
Dynamics of system (2.3) is well-known (e.g., [10]), which is shown as follows. (2)
Theorem 2.2. If μ1 (2.3) converge to E1 .
¯ of (2.3) is globally asymptotically stable in intR2+ . Otherwise, all positive solutions of ¯ v) > 0, equilibrium E12 (u,
Now we consider the predation subsystem consisting of crows and cats. Let v = 0. Then system (2.3) becomes the crow–cat system
du = r1 u(1 − c1 u − b2 w) dt dw = r3 w( − 1 + a2 u − c3 w). dt
(2.5)
Subsystem (2.5) has the same structure as that of (2.3). Denote
a μ(13) = r3 −1 + 2 , uˆ = c1
b2 + c3 a2 − c1 ˆ = , w . a2 b2 + c1 c3 a2 b2 + c1 c3
(2.6)
(3)
ˆ w ˆ ) of (2.5) is globally asymptotically stable in intR2+ . OtherFrom Theorem 2.2, we conclude that if μ1 > 0, equilibrium E13 (u, wise, all positive solutions of (2.5) converge to E1 . For the third subsystem consisting of cuckoos and cats with u = 0, we have dv/dt < 0 and dw/dt < 0. Thus all solutions converge to E0 (0, 0). This is because each of the species cannot survive in the absence of crows. 3. Equilibrium In this section, we show stability of equilibria of system (2.1), which is determined by eigenvalues of the Jacobian matrix. Let f (u, v, w), g(u, v), h(u, v, w) represent the functions on the right-hand side of system (2.1), respectively. Then the Jacobian matrix of system (2.1) takes the form
J(u, v, w) =
fu gu hu
fv gv hv
fw 0 hw
(3.1)
where
fu = r1 (1 − 2c1 u − b1 v − b2 w), fv = −b1 r1 u, fw = −b2 r1 u, gu = a1 r2 v gv = r2 ( − 1 + a1 u − 2c2 v), hu = a2 r3 w, hv = −b3 r3 w, hw = r3 ( − 1 + a2 u − b3 v − 2c3 w). 3.1. Boundary equilibria We have the following boundary equilibria: (a) O(0, 0, 0). The trivial equilibrium O always exists and is a saddle point, where the (v, w)-plane is the stable subspace and the u-axis is the unstable subspace.
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(b) P1 (1/c1 , 0, 0). The semi-trivial equilibrium P1 always exists and the Jacobian matrix J(P1 ) has eigenvalues
a a λ(11) = −r1 , λ(12) = r2 −1 + 1 , λ(13) = r3 −1 + 2 . c1
c1
¯ v¯ , 0). When the boundary equilibrium P12 exists, P12 is globally asymptotically stable on the (u, v)-plane. Thus, we (c) P12 (u, consider its eigenvalue in the w-axis direction
¯ . λ(123) = r3 ( − 1 + a2 u¯ − b3 v) ˆ 0, w ˆ ). When the boundary equilibrium P13 exists, P13 is globally asymptotically stable on the (u, w)-plane. Thus, we (d) P13 (u, consider its eigenvalue in the v-axis direction
λ(132) = r2 ( − 1 + a1 uˆ). 3.2. The positive equilibrium The positive (interior) equilibrium can be obtained from the right-hand side of system (2.1). Let −M be the determinant of the corresponding coefficient matrix, then we have
M = a2 b2 c2 + c1 c2 c3 + a1 b1 c3 − a1 b2 b3 . When M = 0, either there is no positive equilibrium or there is a line segment of positive equilibria in system (2.1). If there is no positive equilibrium, all positive solutions converge to the boundary equilibria which we prove in Proposition 4.2. If there is a line segment of positive equilibria, system (2.1) can be changed to a two-dimensional system [10], which is trivial. Thus, we assume M = 0 in the following discussion. Since M = 0, there is at most one positive equilibrium P ∗ (u∗ , v∗ , w∗ ) of (2.1). It follows from the right-hand side of (2.1) that
b2 c2 + b1 c3 + c2 c3 − b2 b3 M a1 b2 + a1 c3 − a2 b2 − c1 c3 ∗ v = M a b + a c + b3 c1 − a1 b1 − a1 b3 − c1 c2 2 1 2 2 w∗ = . M u∗ =
(3.2)
Denote
a11 =
a2 b2 + c1 c3 , b2 + c3
b31 =
a2 b2 c2 + c1 c2 c3 + a1 b1 c3 , a1 b2
a12 =
a2 b1 + a2 c2 + b3 c1 − c1 c2 b1 + b3 b32 =
b2 c2 + b1 c3 + c2 c3 . b2
Then M > 0 corresponds to b3 < b31 , u∗ > 0 corresponds to b3 < b32 , v∗ > 0 corresponds to a1 > a11 , and w∗ > 0 corresponds to a1 < a12 . From (3.2), we conclude the following result for positivity of P∗ . Proposition 3.1. Assume M = 0. System (2.1) has a positive equilibrium if and only if one of the following conditions is satisfied (i) a11 < a1 < a12 , b3 < min {b31 , b32 }. (ii) a12 < a1 < a11 , b3 > max {b31 , b32 }. The Jacobian matrix of system (2.1) at P∗ is
J(P
∗
)=
−r1 c1 u∗ r2 a1 v∗ r3 a2 w∗
−r1 b1 u∗ −r2 c2 v∗ −r3 b3 w∗
−r1 b2 u∗ 0 . −r3 c3 w∗
(3.3)
The characteristic equation of J(P∗ ) is
λ3 + α1 λ2 + α2 λ + α3 = 0 where
α1 = r1 c1 u∗ + r2 c2 v∗ + r3 c3 w∗ > 0 α2 = (a1 b1 + c1 c2 )r1 r2 u∗ v∗ + (a2 b2 + c1 c3 )r1 r3 u∗ w∗ + c2 c3 r2 r3 v∗ w∗ α3 = (a2 b2 c2 + c1 c2 c3 + a1 b1 c3 − a1 b2 b3 )r1 r2 r3 u∗ v∗ w∗ = Mr1 r2 r3 u∗ v∗ w∗ . Notice that α 3 > 0 implies M > 0 and b3 < b31 . From the Routh–Hurwitz criterion and condition for a Hopf bifurcation by Yu [19], we conclude the following result.
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Theorem 3.2. (i) If
a11 < a1 < a12 , b3 < min{b31 , b32 }, then the positive equilibrium (ii) If
P ∗ (u∗ , v∗ , w∗ )
α1 α2 − α3 > 0
(3.4)
of system (2.1) is asymptotically stable.
a11 < a1 < a12 , b3 < min{b31 , b32 },
α1 α2 − α3 = 0
(3.5)
then a Hopf bifurcation occurs at P∗ in system (2.1). Finally, we give a sufficient condition for the global stability of the positive equilibrium P∗ . ¯ v¯ , 0) are equilibria of system (2.1). Then we have u∗ < u. ¯ Proposition 3.3. Assume that P∗ (u∗ , v∗ , w∗ ) and P12 (u, Proof. Since both P∗ and P12 satisfy the second equation of (2.1), we obtain
v=
−1 + a1 u . c2
Thus, from the first equation of (2.1) and the first equation of (2.3), u∗ and u¯ satisfy the following equations, respectively: 1u u∗ : 1 − c1 u − b1 −1+a − b2 w = 0 with w > 0 c2 1u =0 1 − c1 u − b1 −1+a c2
u¯ :
which implies u∗ < u¯ since w > 0. Theorem 3.4. Assume that there is a positive equilibrium P ∗ (u∗ , v∗ , w∗ ) of system (2.1). If
4a2 b1 c2 c3 ≥ a1 b2 b23
(3.6)
then P∗ is globally asymptotically stable. Proof. Choose a Lyapunov function as follows:
V (u, v, w) = α
u u∗
v w x − u∗ y − v∗ z − w∗ dx + β dy + γ dz x y z v∗ w∗
where α , β and γ are positive constants to be determined. Along any trajectory of system (2.1), we have
dV = r1 α(u − u∗ )(1 − c1 u − b1 v − b2 w) + r2 β(v − v∗ )( − 1 + a1 u − c2 v) dt + r3 γ (w − w∗ )( − 1 + a2 u − b3 v − c3 w) = r1 α(u − u∗ )[−c1 (u − u∗ ) − b1 (v − v∗ ) − b2 (w − w∗ )] + r2 β(v − v∗ )[a1 (u − u∗ ) − c2 (v − v∗ )] + r3 γ (w − w∗ )[a2 (u − u∗ ) − b3 (v − v∗ ) − c3 (w − w∗ )] = −c1 r1 α(u − u∗ )2 − c2 r2 β(v − v∗ )2 − c3 r3 γ (w − w∗ )2 + (a1 r2 β − b1 r1 α)(u − u∗ )(v − v∗ ) + (a2 r3 γ − b2 r1 α)(u − u∗ )(w − w∗ ) − b3 r3 γ (v − v∗ )(w − w∗ ). Choose
α=
1 , r1
β=
b1 , a1 r2
γ=
b2 a2 r3
then we obtain
dV = c1 r1 α(u − u∗ )2 − c2 r2 β(v − v∗ )2 − c3 r3 γ (w − w∗ )2 − b3 r3 γ (v − v∗ )(w − w∗ ) dt
b1 c2 a1 b2 b3 a1 b2 b3 ∗ 2 ∗ 2 ∗ ∗ ∗ = −c1 (u − u ) − (v − v )(w − w ) + (w − w ) 2 (v − v ) + a1 a2 b1 c2 2a2 b1 c2
−
a1 b2 b2 b2 c3 − 22 3 a2 4a2 b1 c2
= −c1 (u − u∗ )2 −
(w − w∗ )2
b2 c2 a1 b2 b3 [(v − v∗ ) + (w − w∗ )]2 − a1 2a2 b1 c2
Thus, if condition (3.6) is satisfied, then
dV dt
≤ 0.
a1 b2 b2 b2 c3 − 22 3 a2 4a2 b1 c2
(w − w∗ )2 .
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We now show that
L=
dV dt
= 0 if and only if u = u∗ , v = v∗ , w = w∗ . Indeed, from
(u, v, w) ∈ R+3 : u = u∗ , w − w∗ = k(v − v∗ ), k = −
dV dt
= 0 we obtain the set
2a2 b1 c2 . a1 b2 b3
Suppose L = {P∗ }. Then L is an invariant line of (2.1) on the plane u = u∗ with slope k. Since k < 0, L and the (u, v)-plane intersect ¯ v¯ , 0). Then we obtain u∗ = u, ¯ which contradicts with Proposition 3.3. Thus L = {P ∗ }, which implies that at equilibrium P12 (u, dV ∗ , v = v∗ , w = w∗ . = 0 if and only if u = u dt Therefore, LaSalle’s Invariance Principle implies that P∗ is globally asymptotically stable. This completes the proof. The main notion of persistence theory is uniform persistence. Consider a dynamical system for n interacting biological species
dxi = xi fi (x1 , x2 , . . . , xn ), i = 1, 2, . . . , n dt
(3.7)
where xi is the density of the ith species. Let x(t ) = (x1 (t ), x2 (t ), . . . , xn (t )) be the solution of (3.7) with positive initial values. System (3.7) is said to be weakly persistent if
lim sup xi (t ) > 0, i = 1, 2, . . . , n t→∞
persistent if
lim inf xi (t ) > 0, i = 1, 2, . . . , n t→∞
uniformly persistent if there is a constant δ 0 > 0 such that for all positive solutions x(t), we have
lim inf xi (t ) ≥ δ0 , i = 1, 2, . . . , n. t→∞
Obviously, a uniformly persistent system is persistent, while a persistent system is weakly persistent. While weak persistence , and uniguarantees survival of each species, persistence guarantees that each solution bounds away from the boundary of R+ 3 . For further discussion form persistence guarantees that all positive solutions uniformly bound away from the boundary of R+ 3 about persistence theory, we refer to Freedman and Moson [8]. 4. Persistence In this section, we exhibit uniform persistence of system (2.1) by applying the Acyclicity Theorem [2,3,9]. To this end, we must ensure that the boundary equilibria do not form a heteroclinic cycle, which is the acyclicity condition in the uniform (2) (3) persistence theorem. Recall that both O and P1 are saddle points when λ1 > 0, λ1 > 0. P12 is unstable in the w-axis direction (3)
(2)
when λ12 > 0, while P13 is unstable in the v-axis direction when λ13 > 0. Therefore, we conclude the main result in this section.
Theorem 4.1. Assume
λ(12) > 0, λ(13) > 0, λ(123) > 0, λ(132) > 0.
(4.1)
Then system (2.1) is uniformly persistent. Now Theorem 4.1 is applied to exhibit properties of system (2.1) when it has one or no positive equilibrium. Denote
a∗2 = a1 +
c3 (a1 − c1 ) ¯ b1 c3 , b3 = . b2 b2
(4.2)
(2)
Then λ13 > 0 can be written as a2 < a∗2 . Proposition 4.2. (i) Let (4.1) hold. Then there is a positive equilibrium P ∗ (u∗ , v∗ , w∗ ) of system (2.1), which satisfies
u∗ − uˆ =
b22 (a∗2 − a2 )(b3 − b¯ 3 ) M(a2 b2 + c1 c3 )
with M > 0.
(4.3)
(ii) If system (2.1) has no positive equilibrium, all positive solutions of (2.1) converge to the boundary of R3+ . Proof. (i) It follows from Theorem 4.1 that system (2.1) is uniformly persistent. The boundedness and uniform persistence of the system (2.1) now guarantee (see [2,12]) that system (2.1) has a positive equilibrium P ∗ (u∗ , v∗ , w∗ ). From (2.6) and (3.2), we obtain
u∗ − uˆ =
(b2 c2 + b1 c3 + c2 c3 − b2 b3 )(a2 b2 + c1 c3 ) − (b2 + c3 )(a2 b2 c2 + c1 c2 c3 + a1 b1 c3 − a1 b2 b3 ) M(a2 b2 + c1 c3 )
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a2 (b3 b22 − b1 b2 c3 ) + b3 [b2 c1 c3 − a1 b2 (b2 + c3 )] − b1 c1 c32 + a1 b1 c3 (b2 + c3 ) M(a2 b2 + c1 c3 ) 2 ∗ ¯ b (a − a2 )(b3 − b3 ) = 2 2 . M(a2 b2 + c1 c3 ) =
(2)
From λ13 > 0 we obtain a2 < a∗2 , which implies a1 b2 + a1 c3 − a2 b2 − c1 c3 > 0. Since v∗ > 0, we have M > 0. (ii) Define the linear function L : R3+ → R3 by L (u, v, w) = (1 − c1 u − b1 v − b2 w, −1 + a1 u − c2 v, −1 + a2 u − b3 v − c3 w). Since system (2.1) admits no positive equilibrium, the convex set S = L (intR3+ ) is disjoint from O(0, 0, 0). The convex theorem for linear operators [10] shows that there is a hyperplane π through O, which is disjoint from the convex set S. Then there is a vector h = (h1 , h2 , h3 ) = 0 orthogonal to π such that h · z < 0 for all z ∈ S. Let (u(t ), v(t ), w(t )) ∈ intR3+ be a solution of (2.1) and denote
V (u, v, w) = log (uh1 vh2 wh3 ) then we have
h2 dv h3 dw h1 du dV + + = h · z < 0, z = L (u, v, w). | = dt (2.1) u dt v dt w dt Thus V(t) is decreasing along each orbit, and no point in intR3+ may belong to the ω-limit set of the orbit by the Lyapunov Theorem [10]. Since all solutions of (2.1) are bounded by Proposition 2.1, they will not blow up to infinity, which implies that they converge to the boundary of R3+ . When condition (4.1) is not satisfied, the following Theorems 4.3 and 4.5 exhibit dynamics of (2.1), which implies extinction of species. Theorem 4.3. Let (u(t ), v(t ), w(t )) ∈ intR3+ be a solution of (2.1). (2)
(i) If λ1
(3)
(ii) If λ1
≤ 0, then limt→+∞ v(t ) = 0. ≤ 0, then limt→+∞ w(t ) = 0.
Proof. Proposition 4.2(ii) is used to prove the result in (i), while a similar proof can be given for (ii). We claim that there is no positive equilibrium of system (2.1). Indeed, suppose that P∗ is a positive equilibrium of (2.1). From (2) the righthand side of the first equation of (2.1), we have c1 u∗ ≤ 1. Since λ1 ≤ 0, we obtain a1 ≤ c1 . Thus a1 u∗ ≤ c1 u∗ ≤ 1. From the righthand side of the second equation of (2.1), we have
−1 + a1 u∗ − c2 v∗ ≤ −c2 v∗ < 0 which forms a contradiction that P∗ is a positive equilibrium. Therefore, it follows from Proposition 4.2 that, all positive solutions converge to the boundary of R3+ . Since O is globally asymptotically stable on the (v, w)-plane and P1 is globally asymptotically stable on the (u, v)-plane, we obtain limt→+∞ v(t ) = 0. From Theorem 4.3 we obtain the following corollary. (2)
Corollary 4.4. If λ1
(3)
≤ 0 and λ1 (2)
Theorem 4.5. Assume λ1 (3)
(2)
(3)
(2)
(3)
(2)
≤ 0, then P1 is globally asymptotically stable. (3)
> 0 and λ1
> 0.
(i) If λ12 < 0 and λ13 > 0, then P12 is globally asymptotically stable.
(ii) If λ12 > 0 and λ13 < 0, then P13 is globally asymptotically stable.
(iii) If λ12 < 0 and λ13 < 0, both P12 and P13 are asymptotically stable. A two-dimensional manifold S2 , which passes through the positive u-axis and positive equilibrium P∗ , divides intR3+ into two regions: one is the basin of attraction of P12 while the other is that of P13 as shown in Fig. 1. Proof. We prove the results by using Proposition 4.2(ii) where it is assumed that there is no positive equilibrium of (2.1). Suppose that system (2.1) admits a positive equilibrium P ∗ (u∗ , v∗ , w∗ ). From (3.2), we have
v∗ M = a2 b2 + c1 c3 − a1 (b2 + c3 ) = −(a2 b2 + c1 c3 )( − 1 + a1 uˆ) a2 b2 + c1 c3 (2) λ13 r2 w∗ M = b3 (a1 − c1 ) − c2 (a2 − c1 ) − b1 (a2 − a1 ) =−
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Bi−stability of equilibria P
12
and P
13
4.5 P13 4
S2
3.5 3
w
2.5 2 1.5 1 0.5 P
12
0 0
5
10 u
15
20
0
15
10
5
20
25
30
v
Fig. 1. Bi-stability of equilibria P12 and P13 . Let a1 = 0.2, a2 = 0.4, b1 = b2 = 0.1, b3 = 0.6, r1 = 1, r2 = r3 = 0.1, ci = 0.1, i = 1, 2, 3. Then we have a2 > a∗2 = 0.3 and b3 > L(a2 ) = 2a2 − 0.3 = 0.5, which implies that both P12 and P13 are asymptotically stable. The two-dimensional manifold S2 , which passes through the positive u-axis, divides intR3+ into two regions: the region below S2 is the basin of attraction of P12 while the region above S2 is that of P13 .
¯ = −(a1 b1 + c1 c2 )( − 1 + a2 u¯ − b3 v) a1 b1 + c1 c2 (3) =− λ12 . r3 (3)
(2)
(i and ii) We give the proof for (i), while a similar one can be given for (ii). Since λ12 < 0 and λ13 > 0, we have v∗ w∗ < 0. Thus there is no positive equilibrium of (2.1). By Proposition 4.2, all positive solutions converge to the boundary of R3+ . Notice that O (resp. P13 ) is globally asymptotically stable on the (v, w)-plane (resp. (u, w)-plane), while P12 is globally asymptotically stable on the (u, v)-plane. Since equilibria O, P1 and P13 are saddle points on R3+ , no positive solution converges to O, P1 or P13 , and all positive solutions converge to P12 , which implies that P12 is globally asymptotically stable in intR3+ . (3)
(iii) Since λ12 < 0 and P12 is globally asymptotically stable on the (u, v)-plane, P12 is asymptotically stable. Similarly, from
(2)
λ13 < 0 we obtain that P13 is asymptotically stable. Let S1 and S2 be the boundaries of attraction basin of P12 and P13 , respectively. We can see that both S1 and S2 include the positive u-axis. On the other hand, since orbits on S1 are bounded and equilibria O, P1 on S1 are saddle points, there exists at least one positive equilibrium on S1 . From the righthand side of (2.1), there is at most one positive equilibrium, which implies that there exists a unique positive equilibrium P∗ ∈ S1 . Similarly, P∗ ∈ S2 . Suppose S1 = S2 . Then the surfaces S1 and S2 surround an open set in intR3+ . Since O and P1 on Si are saddle points, system ¯ by the Acyclicity Theorem, which implies that there is a positive equilibrium in . This is (2.1) is uniformly persistent on
impossible since system (2.1) admits at most one positive equilibrium P ∗ ∈ S1 S2 . Therefore, S1 = S2 = S2 , which implies that 2 3 the two dimensional manifold S divides intR+ into two regions: one is the basin of attraction of P12 while the other is that of P13 . 5. Mutualism In this section we present properties of the parasitism-predation system (2.1) based on Theorems 4.1–4.5. We discuss the effects of parameters a2 and b3 , but similar results can be given for other parameters. Recall that b3 denotes the deterrence effect
S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
673
Bifurcation on the (a ,b )−plane 2
3
b
3
IV: (u,v)/(u,w)
b =L(a ) 3
2
III: (u,w)
I: (u,v)
II: (u,v,w)
0
c
1
a02
a*2
a
2
Fig. 2. Bifurcation diagram on the (a2 , b3 )-plane. Let a1 = 0.2, b1 = b2 = 0.1, r1 = 1, r2 = r3 = 0.1, ci = 0.1, i = 1, 2, 3. Then we have = 0.15, a∗2 = 0.3 and b3 = L(a2 ) = 2a2 − 0.3. The lines a2 = a∗2 and b3 = L(a2 ) divide the first quadrant into four regions. In region I with a2 < a∗2 and b3 > L(a2 ), the label (u, v) means that cats always go extinct. In region II with a02 < a2 < a∗2 and b3 < L(a2 ), the label (u, v, w) means that the three species always coexist. In region III with a2 > a∗2 and b3 < L(a2 ), the label (u, w) means that cuckoos always go extinct. In region IV with a2 > a∗2 and b3 > L(a2 ), the label (u, v)/(u, w) means that either cuckoos or cats go extinct, where the species with higher initial density persists. a02
of cuckoos on cats, and a2 represents the efficiency of cats in converting the predation into fitness, which is determined by cats’ predation ability and the converting coefficient. In the following discussions, we assume ai > c1 , i = 1, 2, which implies that either cuckoos or cats can coexist with crows in the absence of the other by Theorem 4.3. Denote
a02 =
1 −1 + a2 u¯ , L(a2 ) = v¯ u¯
(3)
then λ12 > 0 can be written as b3 < L(a2 ), while the point (a02 , 0) is the intersection of the line b3 = L(a2 ) and the a2 -axis. Since
a02 − c1 =
b1 (a1 − c1 ) >0 b1 + c2
a∗2 − a02 =
(a1 − c1 )(b1 c3 + b2 c2 + c2 c3 ) >0 b2 (b1 + c2 )
we obtain a∗2 > a02 > c1 . As shown in Fig. 2, the lines a2 = a∗2 and b3 = L(a2 ) divide the first quadrant of the (a2 , b3 )-plane into four regions: Region I: a2 < a∗2 , b3 > L(a2 ); Region II: a02 < a2 < a∗2 , b3 < L(a2 ); Region III: a2 > a∗2 , b3 < L(a2 ); Region IV: a2 > a∗2 , b3 > L(a2 ). In region I, cats go extinct by Theorems 4.3 and 4.5(i). In region II, all species persist by Theorem 4.1. In region III, cuckoos go extinct by Theorem 4.5(ii). In region IV, there exists bi-stability of equilibria P12 and P13 by Theorem 4.5(iii). Parameter a2 varies with the cats’ predation ability b2 and efficiency e, and a large a2 ( > a∗2 ) implies an intensive predation b2 or a high efficiency e. Indeed, we have a2 = er1 b2 /r3 by Section 2. Thus, a2 > a02 can be written as e > e0 = r3 a02 /(r1 b2 ), or
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S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
b2 > b02 = r3 a02 /(er1 ), while a2 > a∗2 can be written as e > e∗ = r3 a∗2 /(r1 b2 ). From a2 > a∗2 and (4.2), we obtain
er1 b22 − r3 a1 b2 − c3 r3 (a1 − c1 ) > 0 which implies that a2 > a∗2 can be written as b2 > b∗2 with
b∗2
=
r3 a1 +
r32 a21 + 4c3 r3 (a1 − c1 ) 2er1
.
Therefore, a large a2 ( > a∗2 ) implies an intensive predation b2 ( > b∗2 ) or a high efficiency e( > e∗ ). Theorems 4.1 –4.5 demonstrate basic properties of the crow–cuckoo–cat system. We now offer a biological interpretation of the mathematical results presented in Section 4. First, the three species can coexist when a2 is intermediate and b3 is small. As shown in region II of Fig. 2 and Theorem 4.1, system (2.1) is persistent if a02 < a2 < a∗2 and b3 < L(a2 ). The biological reason is that a weak deterrence (b3 ) will not result in extinction of cats and an intermediate predation/efficiency (a2 ) will not lead to extinction of cuckoos, which allows for the coexistence of the three species. ˆ where u, ¯ uˆ and u∗ are defined Second, when coexisting at an equilibrium, crows and cuckoos are mutualistic only if u¯ > u, ∗ ¯ Thus, if u¯ ≤ u, ˆ we obtain u∗ − uˆ < 0, which in (2.4), (2.6) and (3.2), respectively. As shown in Proposition 3.3, we have u < u. implies that interaction outcomes between crows and cuckoos are not mutualistic. The reason is that u¯ ≤ uˆ means the negative effect of cuckoos on crows is greater than or equal to that of cats. Thus, although cuckoos can decrease the predation of cats, they have a larger negative effect on the growth of crows, which cannot lead to mutualism between them. ˆ interaction outcomes between crows and cuckoos can change from Third, when coexisting at an equilibrium and u¯ > u, parasitism to commensalism and mutualism as b3 increases. To show this change, we need to display that when the positive equilibrium P ∗ (u∗ , v∗ , w∗ ) is asymptotically stable, the sign of u∗ − uˆ changes from negative to positive, where uˆ comes from ˆ 0, w ˆ ). Because of the complexity of u∗ and conditions in Theorem 3.2, we consider a specific case. Fix the equilibrium P13 (u, a1 = 0.2, a2 = 0.25, b1 = b2 = 0.1, r1 = 1, r2 = r3 = 0.1, ci = 0.1 for i = 1, 2, 3 and let b3 < L(a2 ) = 0.2 vary. A direct computation shows that
a02 = 0.15, a∗2 = 0.3, u¯ = u∗ =
10(0.3 − b3 ) , 0.55 − b3
v∗ =
20 10 , v¯ = , L(a2 ) = 2a2 − 0.3 3 3
5 10(4.7 − b3 ) 40 . , w∗ = , uˆ = 0.55 − b3 0.55 − b3 7
Since b3 < 0.2, P ∗ (u∗ , v∗ , w∗ ) is a positive equilibrium. Moreover, since
4a2 b1 c2 c3 = 0.001 > a1 b2 b23 = 0.02b23
as
b3 < 0.2
P∗ is globally asymptotically stable by Theorem 3.4. It follows from
u∗ − uˆ =
100(b3 − 0.1) 7(0.55 − 2b3 )
that interaction outcomes between crows and cuckoos change from parasitism to commensalism and mutualism as b3 increases from b3 < 0.1 to b3 = 0.1 and b3 > 0.1. Numerical simulations in Fig. 3 confirm the change when b3 = 0.05, b3 = 0.1 and b3 = 0.15. The biological reason for the change from parasitism to commensalism and mutualism is as follows. In this parameter region, the predation/efficiency of cats is intermediate. When the deterrence b3 is weak, cuckoos do little contribution to decrease the loss of crows by cats. This implies that the relationship is parasitic since cuckoos depend on crows for persistence. When the deterrence b3 is strong, the crow and cuckoo are mutualistic since a strong deterrence b3 decreases the predation, which decreases the loss of crows by cats compared with that by cuckoos. Fourth, there exists extinction of species when the values of a2 and b3 are unbalanced, i.e., the condition of a02 < a2 < a∗2 , b3 < L(a2 ) is not satisfied. As shown in region I of Fig. 2 and Theorem 4.5(i), when a02 < a2 < a∗2 but b3 > L(a2 ), cats go extinct. The reason is that an extremely strong deterrence b3 prohibits the predation and results in the extinction of cats. Moreover, when c1 < a2 < a02 and b3 > 0, cats go extinct. The reason is that the cats have a very weak predation/efficiency (a2 ) that cannot endure any deterrence (b3 ) from cuckoos. Since cats can survive in the crow-cat system, it is the cuckoo that leads to the extinction. Cuckoos can also be driven into extinction by cats. As shown in region III of Fig. 2 and Theorem 4.5(ii), when a2 > a∗2 but b3 < L(a2 ), cuckoos go extinct. The reason is that a strong predation/efficiency a2 leads to explosive growth of cats and results in the extinction of cuckoos. Since cuckoos can survive in the crow-cuckoo system, it is the cats’ predation that leads to the extinction of cuckoos. Thus, a strong deterrence by a cuckoo not only benefits its host, but also provides itself protection from extinction. Finally, initial densities play a role in survival of species. As shown in region IV of Fig. 2 and Theorem 4.5(iii), P12 and P13 are asymptotically stable when a2 > a∗2 and b3 > L(a2 ). Thus, initial densities in the attraction region of P12 means survival of cuckoos, while those in the attraction region of P13 means survival of cats. The reason is that in the situation where the predation/efficiency is strong and the deterrence is effective, the species with large initial density will be dominant. Since cuckoos and cats can coexist with crows individually, it is the initial density that determines their survival in the three-species system.
S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
(a) Parasitism at b =0.01
(b) Commensalism at b =0.1
3
3
population density
10
10 crow cuckoo cat
8
6
4
4
2
2
0
10
20
30
crow cuckoo cat
8
6
0
40
0
0
population density
(c) Mutualism at b3=0.19 10
8
8
6
6
4
4
2
2
0
50
100 t−time
150
20
40
60
(d) Mutualism at b3=0.8
10
0
675
200
0
0
10
20 t−time
30
40
Fig. 3. Transitions of interaction outcomes between crows and cuckoos from parasitism to commensalism and mutualism when b3 increases. Fix a1 = 0.2, a2 = 0.25, b1 = b2 = 0.1, r1 = 1, r2 = r3 = 0.1, ci = 0.1, i = 1, 2, 3, and let b3 vary. Then we have L(a2 ) = 0.2, which implies that the three species coexist when b3 < 0.2. Here, the density of crows in the crow–cat subsystem is uˆ = 5.7143, which is shown in the red dashed line. Solutions of u, v and w are represented by red, blue and black curves, respectively. (a) When b3 = 0.01 < 0.1, the outcomes are parasitism. (b) When b3 = 0.1, the outcomes are commensalism. (c) When b3 = 0.19 > 0.1, the outcomes are mutualism. (d) When b3 = 0.8 > 0.2, cats are driven into extinction and the outcomes are mutualism. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
6. Discussion In this paper, we consider mutualism in a parasitism-predation system consisting of crows, cuckoos and cats. Dynamics of the three-species model demonstrate the mechanism by which the crow and cuckoo can be mutualistic in the presence of cats, as shown in Theorems 4.1–4.5. Results in this work not only explain biological observation by Canestrari et al. [4], but also predict situations that have not been observed by Canestrari et al. [4]. We focus on parameter a2 , which represents both the predation degree and efficiency of cats in converting the predation into fitness, as shown in Section 2. For simplicity, we assume u¯ > uˆ and the system persists at a steady state P ∗ (u∗ , v∗ , w∗ ). Recall that u¯ and uˆ denote crows’ density in the crow–cuckoo subsystem and crow-cat subsystem, respectively. (A) When a2 is small (i.e., c1 < a2 < a02 ), Theorem 4.5(i) exhibits that cats will be driven into extinction by cuckoos, which implies that crows approach a high density u¯ ( > uˆ). Thus, the crow and cuckoo are mutualistic in this case. (B) Assume that a2 is intermediate (i.e., a02 < a2 < a∗2 ). If the negative effect b3 of cuckoos on cats is small (b3 < L(a2 )), Theorem 4.1 displays that the three species coexist and crows will approach a high density u∗ ( > uˆ) by (4.3). Thus, the crow and cuckoo are mutualistic, which explains the observation by Canestrari et al. [4]. If b3 > L(a2 ), Theorem 4.5(i) exhibits that cats will be driven into extinction by cuckoos, which implies that the crow and cuckoo are mutualistic as shown in (A). (C) Assume that a2 is large (i.e., a2 > a∗2 ). If b3 < L(a2 ), Theorem 4.5(ii) displays that cuckoos will be driven into extinction by cats, which implies that crows approach the ˆ Thus, the crow and cuckoo are neutralistic. If b3 > L(a2 ), Theorem 4.5(iii) displays that either cuckoos or cats go extinct. density u. ˆ When cats go extinct, the When cuckoos go extinct, the crow and cuckoo are neutralistic since crows approach the density u. crow and cuckoo are mutualistic since crows approach a high density u¯ ( > uˆ). Periodic oscillations can occur in the crow–cuckoo–cat system. As shown in Theorem 3.2(ii), there exists a periodic solution of system (2.1) when a Hopf bifurcation appears. For example, let r1 = r2 = r3 = 1, a1 = 0.4, a2 = 0.7, b1 = b2 = 0.1, b3 = 0.00001, c1 = 0.01, c2 = 0.1, c3 = 0.00001. Then the solution of system (2.1) with initial value (0.3,0.3,0.3) is in periodic oscillation, as shown in Fig. 4d.
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S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
(a) b =0.5 with initial value (0.3,0.8,0.1)
(b) b =5.5 with initial value (0.3,0.8,0.1)
3
population density
10
3
10 crow cuckoo cat
8 6
6
4
4
2
2
0
0
10
20
30
crow cuckoo cat
8
40
0
0
population density
(c) b3=5.5 with initial value (0.3,0.1,0.8)
10
20
30
40
(d) Hopf bifurcation in system (2.1)
10
60
8
50 40
6 30 4 20 2 0
10 0
10
20 t−time
30
40
0
0
200
400 t−time
600
800
Fig. 4. Extinction of species and Hopf bifurcation. In (a)–(c), fix a1 = 0.2, a2 = 0.4, b1 = b2 = 0.1, r1 = 1, r2 = r3 = 0.1, ci = 0.1 for i = 1, 2, 3, and let b3 vary. Then we have a2 > a∗2 = 0.3, which implies that P13 is asymptotically stable. Here, L(a2 ) = 0.5. Solutions of u, v and w are represented by red, blue and black curves, respectively. (a) When b3 = 0.5, cuckoos go extinct. (b) When b3 = 5.5 with initial value (0.3,0.8,0.1), cats are driven into extinction by cuckoos. (c) When b3 = 5.5 with initial value (0.3,0.1,0.8), cuckoos are driven into extinction by cats. (d) Hopf bifurcation. Let r1 = r2 = r3 = 1, a1 = 0.4, a2 = 0.7, b1 = b2 = 0.1, b3 = 0.00001, c1 = 0.01, c2 = 0.1, c3 = 0.00001. The solution with initial value (0.3,0.3,0.3) is periodic. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The biological observations by Canestrari et al. [4] support the crow–cuckoo–cat model in this work. From the observations by Canestrari et al. [4], we obtain the approximated parameter values in model (2.1) on MatLab by least square method of linear regression: r1 = 6.5459, r2 = 0.0816, r3 = 0.0001, r2 a1 = 0.08 r1 b2 = 0.1, r1 b3 = 0.0238 r3 a2 = 0.05, r1 b1 = 0.1, r1 c1 = 0.5919, r2 c2 = 0.1562, r3 c3 = 0.0033 with initial value (3.0084, 0.5741, 47.5491). Numerical simulations in Fig. 5 show that our model fits the biological observation well. Moreover, numerical simulations in Fig. 5 show that u∗ = 3.5018 > uˆ = 3.1353. Thus the crow and cuckoo are mutualistic, which is consistent with both Theorem 4.1 and the observation by Canestrari et al. [4]. The assumption (3.6) is consistent with the positivity of P∗ , which can be shown by simulations in Fig. 3a. Here, we have r1 = 1, r2 = r3 = 0.1, a1 = 0.2, a2 = 0.25, b1 = b2 = 0.1, b3 = 0.01, ci = 0.1, i = 1, 2, 3. Numerical simulations display that there is a positive equilibrium P∗ . Moreover, a direction computation exhibits that
4a2 b1 c2 c3 = 10−2 > 2 × 10−5 = a1 b2 b23 which implies that there is no contradiction between the assumption (3.6) and the positivity of P∗ . Functional responses in the present model are linear. If more general functional responses (e.g., saturated responses) are used, there may exist more complicated dynamics such as chaos in the system, which will be studied in our further study. Acknowledgements We are grateful to the three anonymous reviewers for their careful reading, helpful comments and suggestions that really helped us to improve the presentation of the paper. This work was supported by Student Research Training Program of Zhejiang University, NSF of Guangdong (S2012010010320) and NSF of People’s Republic of China (11571382).
S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
677
Fitting the real data with model (2.3) 50
45
40
population density
35
30 crow cuckoo cat
25
20
15
10
5
0
0
5
10
15 t−time
20
25
30
Fig. 5. Fitting the empirical observation with model (2.1). The data observed by Canestrari et al. [4] is shown in red ∗, while solutions of u, v and w are represented by red, blue and black curves, respectively. Let r1 = 6.5459, r2 = 0.0816, r3 = 0.0001, r2 a1 = 0.08, r3 a2 = 0.05, r1 b1 = r1 b2 = 0.1, r1 b3 = 0.0238, r1 c1 = 0.5919, r2 c2 = 0.1562, r3 c3 = 0.0033 with initial value (3.0084, 0.5741, 47.5491). Numerical simulations show that our model fits the empirical observation well and the crow and cuckoo are mutualistic in the presence of cats. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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R.A. Armstrong, R. McGehee, Competitive exclusion, Am. Nat. 115 (1980) 151–170. G.J. Butler, H.I. Freedman, P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Sco. 96 (1986) 425–430. G.J. Butler, P. Waltman, Persistence in dynamical systems, J. Differ. Equ. 63 (1986) 255–263. D. Canestrari, D. Bolopo, T.C.J. Turlings, G. Roder, J.M. Marcos, V. Baglione, From parasitism to mutualism: unexpected interactions between a cuckoo and its host, Science 343 (2014) 1350–1352. R.S. Cantrell, C. Cosner, S. Ruan, Intraspecific interference and consumer-resource dynamics, Discrete Conti. Dyn. Syst.-B 4 (2004) 527–546. C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol. 58 (1996) 207–246. H.I. Freedman, P. Waltman, Persistence in models of three interacting predator–prey populations, Math. Biosci. 68 (1984) 213–231. H.I. Freedman, P. Moson, Persistence definitions and their connections, Proc. Amer. Math. Soc. 109 (1990) 1025–1033. H.I. Freedman, S. Ruan, M. Tang, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Differ. Equ. 6 (1994) 583–600. J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998. S.B. Hsu, S. Ruan, T.H. Yang, On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response, Discrete Conti. Dyn. Syst.-B 18 (9) (2013) 2331–2353. V. Hutson, K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci. 111 (1992) 1–71. B. Lisena, Asymptotic behavior in periodic three species predator–prey systems, Annali di Matematica Pura ed Applicata 186 (1) (2007) 85–98. A.J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. H.L. Smith, H.R. Thieme, Dynamical Systems and Population Persistence, Am. Math. Soc., 2011. Y. Wang, Wu H., Invasibility of nectarless flowers in plant-pollinator systems, Bull. Math. Biol. 75 (2013) 1138–1156. Y. Wang, D.L. DeAngelis, J.N. Holland, Dynamics of an ant–plant–pollinator model, Commun. Non. Sci. Numer. Simul. 20 (2015) 950–964. V. Volterra, Variations and fluctuations of a number of individuals in animal species living together, Mem. Acad. Lincei. 2 (1926) 31–113. P. Yu, Closed-form conditions of bifurcation points for general differential equations, Int J Bifur. Chaos. Appl. Sci. Eng. 15 (2005) 1467–1483.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Transition of interactions between a cuckoo and its host: Fluctuating between parasitism and mutualism Shikun Wang a, Yuanshi Wang b,∗ a b
School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, PR China School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, PR China
a r t i c l e
i n f o
Keywords: Cooperation Persistence Coexistence Stability Bifurcation
a b s t r a c t This paper considers crow–cuckoo–cat systems in which crows are the host, cuckoos are the parasite, cats are the predator of crow nests, and cuckoo chicks in the nests reveal a mix of caustic and repulsive compounds that repel and deter cats. An over 16-year observation on the crow–cuckoo–cat system shows that the parasite and its host can be mutualistic in the presence of predators (Canestrari et al., 2014). In order to exhibit mechanisms by which the mutualism occurs, a crow–cuckoo–cat model is formed in this work. Global dynamics of the model demonstrate that cats’ predation ability and converting efficiency are crucial to the occurrence of mutualism. When the predation/efficiency is low, cats would be driven into extinction by cuckoos. When the predation/efficiency is intermediate, the three species coexist and interaction outcomes between crows and cuckoos change from parasitism to commensalism and mutualism as the deterrence effect increases. When the predation/efficiency is high, cats would drive cuckoos into extinction if the deterrence is weak. If the deterrence is strong, either cuckoos or cats go extinct, where the species with higher initial density persists. Therefore, results in this work show the empirical observation by Canestrari et al. (2014), and predict situations that have not been observed by Canestrari et al. (2014). Numerical simulations display that the crow–cuckoo–cat model fits the observation well. © 2015 Elsevier Inc. All rights reserved.
1. Introduction An over 16-year observation on carrion crows (Corvus corone corone) and the great spotted cuckoo (Clamator glandarius) shows that a parasite (cuckoo) can provide a benefit to its host (crow) by deterring crow nest predators [4]. The cuckoo is called a parasite of the crow since it lays eggs in the nest of the crow, which breeds the unrelated chicks and suffers the loss of its brood. Canestrari et al. [4] studied crow nests in northern Spain and found that the nests parasitized by cuckoos are more successful in crows’ raising their own chicks than non-parasitized nests. This is due to predator repellence by a malodorous cloacal secretion that parasitic chicks release. The cuckoo secretion is a mixture of caustic and repulsive compounds, which are known to repel crow nest predators such as corvids, raptor birds and quasi-feral cats. A cat is called quasi-feral if it is free-ranging and hunts year-round but could be attracted with food. For convenience, we use cats to represent the predators in this work. The long-term observation by Canestrari et al. [4] displays that although cuckoos decreased host reproductive success during seasons with low nest predation, parasitized nests produced more fledglings than non-parasitized nests during seasons with
∗
Corresponding author. Tel.: +86 20 8403 4848; fax: +86 20 8403 7978. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.amc.2015.10.067 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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high nest predation. Thus, interaction outcomes between the crow and cuckoo fluctuate between parasitism and mutualism each season, depending on the intensity of predation pressure. While the novel mutualism between the parasite and its host is observed, mechanisms by which the mutualism can occur need to be demonstrated. Parasitism/predation systems of two species have been studied by a series of significant works. Volterra [18] established a predator–prey model to describe interactions between the predator fish and prey fish in Adriatic during the First World War. Functional responses in the model are linear and all positive solutions (except the equilibrium) are shown to be periodic. Since the time averages of all periodic solutions correspond to components of the unique positive equilibrium, dynamics of the model well characterize the mechanism by which the amount of predator fish during the First World War was considerably higher than those in the years before. The model is also called the Lotka–Volterra predator–prey model because of the important contribution of Lotka [14]. Moreover, the model is extended to describe the predation system where there exists intraspecific competition in each species. The extended model is structurally stable and all of its positive solutions converge to an equilibrium (see for example, [10]). The two-species model is expanded to describe predation systems with multiple species. Freedman and Waltman [7] considered a general predator–prey system of three species, in which two predators have a common prey. A general mathematical criterion is given for persistence of the three-species system. Cantrell et al. [5] studied a specific two-consumer and one-resource system in which there is no interaction between the consumers. Theoretical analysis and numerical simulations showed that the two consumers can coexist upon the single resource at steady states or in periodic oscillations as the environmental condition varies. Lisena [13] considered a two-predator and one-prey system with periodic coefficients, in which there is competition between the predators. Suitable average conditions are given to extinction of the competitively inferior species. For more relevant works, see Armstrong and McGehee [1], Smith and Thieme [15], Hsu et al. [11], Wang and Wu [16], Wang et al. [17], etc. While these works focus on coexistence of consumers, there is little work on parasitism-predation models in which the parasite and its host can be mutualistic in the presence of predators. Thus, modelling the crow–cuckoo–cat system and exhibiting mechanisms by which the mutualism between the crow and cuckoo can occur is necessary. Motivated by the empirical observation and Lotka–Volterra predator–prey model, we expand the model to describe the parasitism-predation system consisting of crows, cuckoos and cats. In the system, crows are the host, cuckoos are the parasites, cats are the predators of crows, and cuckoo chicks reveal a mix of caustic and repulsive compounds that repel and deter cats. Global dynamics of the crow–cuckoo–cat model demonstrates that cats’ predation ability and converting efficiency are crucial to the occurrence of the mutualism between crows and cuckoos. When the predation/efficiency is low, cats would be driven into extinction by cuckoos. When the predation/efficiency is intermediate, the three species coexist and interaction outcomes between crows and cuckoos change from parasitism to commensalism and mutualism as the deterrence effect increases. When the predation/efficiency is high, cats would drive cuckoos into extinction if the deterrence is weak. If the deterrence is strong, either cuckoos or cats go extinct, where the species with higher initial density persists. Therefore, results in this work show the empirical observation by Canestrari et al. [4], and predict situations that have not been observed by Canestrari et al. [4]. Simulations illustrate the results. The paper is organized as follows. The crow–cuckoo–cat model is formed in Section 2. In Section 3, we consider stability of equilibria and in Section 4, we show persistence of the system. We exhibit occurrence of mutualism in Section 5, and a discussion of the results is in Section 6. 2. The crow–cuckoo–cat model In this section, a three-species model is formed to characterize the crow–cuckoo–cat system. Since cuckoo chicks consume food provided by adult crows, the crow-cuckoo interaction is parasitism, which can be described by a parasitism model. On the other hand, since cats kill and eat crows, the crow–cat interaction is predation, which can be depicted by a predator–prey model. Then the two models are combined to form a crow–cuckoo–cat model, and boundedness of solutions of the three-species model is shown. In the crow–cuckoo interaction, immature cuckoos are bred by crows, which results in a loss of crows in their own brood. Thus the crow-cuckoo interaction is parasitic and we apply the Volterra type model to depict the interaction
u + β1 v du = r1 u 1 − dt K dv = dt
v( − r2 + α1 u − γ1 v)
where variables u and v denote population densities of crows and cuckoos, respectively. Parameter r1 represents the intrinsic growth rate of crows, while K is the carrying capacity. Parameter β 1 denotes the quantity of resources consumed by a cuckoo, and α 1 is the efficiency of the cuckoo in converting the consumption into fitness. Parameter r2 is the mortality rate of cuckoos, while γ 1 represents the intraspecific competition. In the crow–cat interaction, cats are predators and crows are their prey, which can be described by
u du − β2 uw = r1 u 1 − dt K dw = w( − r3 + α2 u − γ2 w) dt
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where variables u and w represent population densities of crows and cats, respectively. Parameter β 2 represents the rate of crows killed by a cat, which denotes predation ability of the cat. Parameter α2 = eβ2 with 0 < e < 1 represents the efficiency of cats in converting the predation into fitness, which is determined by cats’ predation ability (β 2 ) and the converting coefficient (e). Parameter r3 is the mortality rate of cats. In the crow–cuckoo–cat interactions, an immature cuckoo produces a mix of caustic and repulsive compounds, which deters the predation of cats on the crows’ nest. Thus, the cuckoo has a negative effect on the growth of cats. Similar to the Lotka–Volterra predator–prey model, we apply a linear functional response to depict the effect by subtracting a term proportional to v. Moreover, it has been observed that cats do not attack cuckoos with the compounds (page 1351, [4]). By combining the above crow–cuckoo and crow–cat models, we obtain a crow–cuckoo–cat model
u + β1 v du = r1 u 1 − dt K
− β2 uw
dv = v( − r2 + α1 u − γ1 v) dt dw = w( − r3 + α2 u − β3 v − γ2 w) dt For simplicity, we rewrite the crow–cuckoo–cat model by
du = r1 u(1 − c1 u − b1 v − b2 w) dt dv = r2 v( − 1 + a1 u − c2 v) dt dw = r3 w( − 1 + a2 u − b3 v − c3 w) dt
(2.1)
with initial conditions
u(0) ≥ 0,
v(0) ≥ 0, w(0) ≥ 0.
(2.2)
Parameter b3 represents the negative effect of cuckoos on the growth of cats, and all parameters in system (2.1) are positive, which satisfy the following relations
a1 =
α1 r2
, a2 =
α2 r3
, b1 =
β1 K
, b2 =
β2 r1
, b3 =
β3 r3
, c1 =
1 γ1 γ2 , c3 = . , c2 = K r2 r3
We can see that the solutions to the initial value problem (2.1) and (2.2) are nonnegative. We also have the following result on boundedness of the solutions. Proposition 2.1. Solutions of system (2.1) are bounded. Proof. From the first equation of system (2.1), we have
du ≤ r1 u(1 − c1 u) dt so that the comparison principle [6] implies that
lim sup u(t ) ≤ K1 , K1 = t→∞
1 . c1
Thus, for 0 > 0 small, we have u(t ) ≤ 0 + K1 when t is sufficiently large. From the second equation in system (2.1), we have
dv ≤ r2 v(Kˆ2 − c2 v), Kˆ2 = −1 + a1 (0 + K1 ). dt If Kˆ2 ≤ 0, then limt→∞ v(t ) = 0. If Kˆ2 > 0, then
lim sup v(t ) ≤ K2 , K2 = t→∞
Kˆ2 . c2
Thus, we conclude lim supt→∞ v(t ) ≤ |K2 |. Similarly, we obtain
lim sup w(t ) ≤ |K3 |, K3 = t→∞
−1 + a2 (0 + K1 ) . c3
Therefore, solutions of system (2.1) are bounded.
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There are three two-species subsystems to be considered. In this section, we cite some known results on the dynamics of the two-species subsystems. We first consider the parasitism subsystem consisting of crows and cuckoos. Let w = 0. Then system (2.3) becomes the crow-cuckoo system
du = r1 u(1 − c1 u − b1 v) dt dv = r2 v( − 1 + a1 u − c2 v). dt
(2.3)
This is the Lotka–Volterra predator–prey system. There are two equilibria on the axes, namely E0 = (0, 0) and E1 = (1/c1 , 0). E0 is a saddle point with eigenvalues
μ(01) = r1 , μ(02) = −r2 . E1 has eigenvalues
a μ(11) = −r1 , μ(12) = r2 −1 + 1 . c1
(2)
¯ of (2.3) exists if μ1 ¯ v) The positive equilibrium E12 (u,
u¯ =
> 0 with
a1 − c1 b1 + c2 , v¯ = . a1 b1 + c1 c2 a1 b1 + c1 c2
(2.4)
Dynamics of system (2.3) is well-known (e.g., [10]), which is shown as follows. (2)
Theorem 2.2. If μ1 (2.3) converge to E1 .
¯ of (2.3) is globally asymptotically stable in intR2+ . Otherwise, all positive solutions of ¯ v) > 0, equilibrium E12 (u,
Now we consider the predation subsystem consisting of crows and cats. Let v = 0. Then system (2.3) becomes the crow–cat system
du = r1 u(1 − c1 u − b2 w) dt dw = r3 w( − 1 + a2 u − c3 w). dt
(2.5)
Subsystem (2.5) has the same structure as that of (2.3). Denote
a μ(13) = r3 −1 + 2 , uˆ = c1
b2 + c3 a2 − c1 ˆ = , w . a2 b2 + c1 c3 a2 b2 + c1 c3
(2.6)
(3)
ˆ w ˆ ) of (2.5) is globally asymptotically stable in intR2+ . OtherFrom Theorem 2.2, we conclude that if μ1 > 0, equilibrium E13 (u, wise, all positive solutions of (2.5) converge to E1 . For the third subsystem consisting of cuckoos and cats with u = 0, we have dv/dt < 0 and dw/dt < 0. Thus all solutions converge to E0 (0, 0). This is because each of the species cannot survive in the absence of crows. 3. Equilibrium In this section, we show stability of equilibria of system (2.1), which is determined by eigenvalues of the Jacobian matrix. Let f (u, v, w), g(u, v), h(u, v, w) represent the functions on the right-hand side of system (2.1), respectively. Then the Jacobian matrix of system (2.1) takes the form
J(u, v, w) =
fu gu hu
fv gv hv
fw 0 hw
(3.1)
where
fu = r1 (1 − 2c1 u − b1 v − b2 w), fv = −b1 r1 u, fw = −b2 r1 u, gu = a1 r2 v gv = r2 ( − 1 + a1 u − 2c2 v), hu = a2 r3 w, hv = −b3 r3 w, hw = r3 ( − 1 + a2 u − b3 v − 2c3 w). 3.1. Boundary equilibria We have the following boundary equilibria: (a) O(0, 0, 0). The trivial equilibrium O always exists and is a saddle point, where the (v, w)-plane is the stable subspace and the u-axis is the unstable subspace.
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(b) P1 (1/c1 , 0, 0). The semi-trivial equilibrium P1 always exists and the Jacobian matrix J(P1 ) has eigenvalues
a a λ(11) = −r1 , λ(12) = r2 −1 + 1 , λ(13) = r3 −1 + 2 . c1
c1
¯ v¯ , 0). When the boundary equilibrium P12 exists, P12 is globally asymptotically stable on the (u, v)-plane. Thus, we (c) P12 (u, consider its eigenvalue in the w-axis direction
¯ . λ(123) = r3 ( − 1 + a2 u¯ − b3 v) ˆ 0, w ˆ ). When the boundary equilibrium P13 exists, P13 is globally asymptotically stable on the (u, w)-plane. Thus, we (d) P13 (u, consider its eigenvalue in the v-axis direction
λ(132) = r2 ( − 1 + a1 uˆ). 3.2. The positive equilibrium The positive (interior) equilibrium can be obtained from the right-hand side of system (2.1). Let −M be the determinant of the corresponding coefficient matrix, then we have
M = a2 b2 c2 + c1 c2 c3 + a1 b1 c3 − a1 b2 b3 . When M = 0, either there is no positive equilibrium or there is a line segment of positive equilibria in system (2.1). If there is no positive equilibrium, all positive solutions converge to the boundary equilibria which we prove in Proposition 4.2. If there is a line segment of positive equilibria, system (2.1) can be changed to a two-dimensional system [10], which is trivial. Thus, we assume M = 0 in the following discussion. Since M = 0, there is at most one positive equilibrium P ∗ (u∗ , v∗ , w∗ ) of (2.1). It follows from the right-hand side of (2.1) that
b2 c2 + b1 c3 + c2 c3 − b2 b3 M a1 b2 + a1 c3 − a2 b2 − c1 c3 ∗ v = M a b + a c + b3 c1 − a1 b1 − a1 b3 − c1 c2 2 1 2 2 w∗ = . M u∗ =
(3.2)
Denote
a11 =
a2 b2 + c1 c3 , b2 + c3
b31 =
a2 b2 c2 + c1 c2 c3 + a1 b1 c3 , a1 b2
a12 =
a2 b1 + a2 c2 + b3 c1 − c1 c2 b1 + b3 b32 =
b2 c2 + b1 c3 + c2 c3 . b2
Then M > 0 corresponds to b3 < b31 , u∗ > 0 corresponds to b3 < b32 , v∗ > 0 corresponds to a1 > a11 , and w∗ > 0 corresponds to a1 < a12 . From (3.2), we conclude the following result for positivity of P∗ . Proposition 3.1. Assume M = 0. System (2.1) has a positive equilibrium if and only if one of the following conditions is satisfied (i) a11 < a1 < a12 , b3 < min {b31 , b32 }. (ii) a12 < a1 < a11 , b3 > max {b31 , b32 }. The Jacobian matrix of system (2.1) at P∗ is
J(P
∗
)=
−r1 c1 u∗ r2 a1 v∗ r3 a2 w∗
−r1 b1 u∗ −r2 c2 v∗ −r3 b3 w∗
−r1 b2 u∗ 0 . −r3 c3 w∗
(3.3)
The characteristic equation of J(P∗ ) is
λ3 + α1 λ2 + α2 λ + α3 = 0 where
α1 = r1 c1 u∗ + r2 c2 v∗ + r3 c3 w∗ > 0 α2 = (a1 b1 + c1 c2 )r1 r2 u∗ v∗ + (a2 b2 + c1 c3 )r1 r3 u∗ w∗ + c2 c3 r2 r3 v∗ w∗ α3 = (a2 b2 c2 + c1 c2 c3 + a1 b1 c3 − a1 b2 b3 )r1 r2 r3 u∗ v∗ w∗ = Mr1 r2 r3 u∗ v∗ w∗ . Notice that α 3 > 0 implies M > 0 and b3 < b31 . From the Routh–Hurwitz criterion and condition for a Hopf bifurcation by Yu [19], we conclude the following result.
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Theorem 3.2. (i) If
a11 < a1 < a12 , b3 < min{b31 , b32 }, then the positive equilibrium (ii) If
P ∗ (u∗ , v∗ , w∗ )
α1 α2 − α3 > 0
(3.4)
of system (2.1) is asymptotically stable.
a11 < a1 < a12 , b3 < min{b31 , b32 },
α1 α2 − α3 = 0
(3.5)
then a Hopf bifurcation occurs at P∗ in system (2.1). Finally, we give a sufficient condition for the global stability of the positive equilibrium P∗ . ¯ v¯ , 0) are equilibria of system (2.1). Then we have u∗ < u. ¯ Proposition 3.3. Assume that P∗ (u∗ , v∗ , w∗ ) and P12 (u, Proof. Since both P∗ and P12 satisfy the second equation of (2.1), we obtain
v=
−1 + a1 u . c2
Thus, from the first equation of (2.1) and the first equation of (2.3), u∗ and u¯ satisfy the following equations, respectively: 1u u∗ : 1 − c1 u − b1 −1+a − b2 w = 0 with w > 0 c2 1u =0 1 − c1 u − b1 −1+a c2
u¯ :
which implies u∗ < u¯ since w > 0. Theorem 3.4. Assume that there is a positive equilibrium P ∗ (u∗ , v∗ , w∗ ) of system (2.1). If
4a2 b1 c2 c3 ≥ a1 b2 b23
(3.6)
then P∗ is globally asymptotically stable. Proof. Choose a Lyapunov function as follows:
V (u, v, w) = α
u u∗
v w x − u∗ y − v∗ z − w∗ dx + β dy + γ dz x y z v∗ w∗
where α , β and γ are positive constants to be determined. Along any trajectory of system (2.1), we have
dV = r1 α(u − u∗ )(1 − c1 u − b1 v − b2 w) + r2 β(v − v∗ )( − 1 + a1 u − c2 v) dt + r3 γ (w − w∗ )( − 1 + a2 u − b3 v − c3 w) = r1 α(u − u∗ )[−c1 (u − u∗ ) − b1 (v − v∗ ) − b2 (w − w∗ )] + r2 β(v − v∗ )[a1 (u − u∗ ) − c2 (v − v∗ )] + r3 γ (w − w∗ )[a2 (u − u∗ ) − b3 (v − v∗ ) − c3 (w − w∗ )] = −c1 r1 α(u − u∗ )2 − c2 r2 β(v − v∗ )2 − c3 r3 γ (w − w∗ )2 + (a1 r2 β − b1 r1 α)(u − u∗ )(v − v∗ ) + (a2 r3 γ − b2 r1 α)(u − u∗ )(w − w∗ ) − b3 r3 γ (v − v∗ )(w − w∗ ). Choose
α=
1 , r1
β=
b1 , a1 r2
γ=
b2 a2 r3
then we obtain
dV = c1 r1 α(u − u∗ )2 − c2 r2 β(v − v∗ )2 − c3 r3 γ (w − w∗ )2 − b3 r3 γ (v − v∗ )(w − w∗ ) dt
b1 c2 a1 b2 b3 a1 b2 b3 ∗ 2 ∗ 2 ∗ ∗ ∗ = −c1 (u − u ) − (v − v )(w − w ) + (w − w ) 2 (v − v ) + a1 a2 b1 c2 2a2 b1 c2
−
a1 b2 b2 b2 c3 − 22 3 a2 4a2 b1 c2
= −c1 (u − u∗ )2 −
(w − w∗ )2
b2 c2 a1 b2 b3 [(v − v∗ ) + (w − w∗ )]2 − a1 2a2 b1 c2
Thus, if condition (3.6) is satisfied, then
dV dt
≤ 0.
a1 b2 b2 b2 c3 − 22 3 a2 4a2 b1 c2
(w − w∗ )2 .
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S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
We now show that
L=
dV dt
= 0 if and only if u = u∗ , v = v∗ , w = w∗ . Indeed, from
(u, v, w) ∈ R+3 : u = u∗ , w − w∗ = k(v − v∗ ), k = −
dV dt
= 0 we obtain the set
2a2 b1 c2 . a1 b2 b3
Suppose L = {P∗ }. Then L is an invariant line of (2.1) on the plane u = u∗ with slope k. Since k < 0, L and the (u, v)-plane intersect ¯ v¯ , 0). Then we obtain u∗ = u, ¯ which contradicts with Proposition 3.3. Thus L = {P ∗ }, which implies that at equilibrium P12 (u, dV ∗ , v = v∗ , w = w∗ . = 0 if and only if u = u dt Therefore, LaSalle’s Invariance Principle implies that P∗ is globally asymptotically stable. This completes the proof. The main notion of persistence theory is uniform persistence. Consider a dynamical system for n interacting biological species
dxi = xi fi (x1 , x2 , . . . , xn ), i = 1, 2, . . . , n dt
(3.7)
where xi is the density of the ith species. Let x(t ) = (x1 (t ), x2 (t ), . . . , xn (t )) be the solution of (3.7) with positive initial values. System (3.7) is said to be weakly persistent if
lim sup xi (t ) > 0, i = 1, 2, . . . , n t→∞
persistent if
lim inf xi (t ) > 0, i = 1, 2, . . . , n t→∞
uniformly persistent if there is a constant δ 0 > 0 such that for all positive solutions x(t), we have
lim inf xi (t ) ≥ δ0 , i = 1, 2, . . . , n. t→∞
Obviously, a uniformly persistent system is persistent, while a persistent system is weakly persistent. While weak persistence , and uniguarantees survival of each species, persistence guarantees that each solution bounds away from the boundary of R+ 3 . For further discussion form persistence guarantees that all positive solutions uniformly bound away from the boundary of R+ 3 about persistence theory, we refer to Freedman and Moson [8]. 4. Persistence In this section, we exhibit uniform persistence of system (2.1) by applying the Acyclicity Theorem [2,3,9]. To this end, we must ensure that the boundary equilibria do not form a heteroclinic cycle, which is the acyclicity condition in the uniform (2) (3) persistence theorem. Recall that both O and P1 are saddle points when λ1 > 0, λ1 > 0. P12 is unstable in the w-axis direction (3)
(2)
when λ12 > 0, while P13 is unstable in the v-axis direction when λ13 > 0. Therefore, we conclude the main result in this section.
Theorem 4.1. Assume
λ(12) > 0, λ(13) > 0, λ(123) > 0, λ(132) > 0.
(4.1)
Then system (2.1) is uniformly persistent. Now Theorem 4.1 is applied to exhibit properties of system (2.1) when it has one or no positive equilibrium. Denote
a∗2 = a1 +
c3 (a1 − c1 ) ¯ b1 c3 , b3 = . b2 b2
(4.2)
(2)
Then λ13 > 0 can be written as a2 < a∗2 . Proposition 4.2. (i) Let (4.1) hold. Then there is a positive equilibrium P ∗ (u∗ , v∗ , w∗ ) of system (2.1), which satisfies
u∗ − uˆ =
b22 (a∗2 − a2 )(b3 − b¯ 3 ) M(a2 b2 + c1 c3 )
with M > 0.
(4.3)
(ii) If system (2.1) has no positive equilibrium, all positive solutions of (2.1) converge to the boundary of R3+ . Proof. (i) It follows from Theorem 4.1 that system (2.1) is uniformly persistent. The boundedness and uniform persistence of the system (2.1) now guarantee (see [2,12]) that system (2.1) has a positive equilibrium P ∗ (u∗ , v∗ , w∗ ). From (2.6) and (3.2), we obtain
u∗ − uˆ =
(b2 c2 + b1 c3 + c2 c3 − b2 b3 )(a2 b2 + c1 c3 ) − (b2 + c3 )(a2 b2 c2 + c1 c2 c3 + a1 b1 c3 − a1 b2 b3 ) M(a2 b2 + c1 c3 )
S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
671
a2 (b3 b22 − b1 b2 c3 ) + b3 [b2 c1 c3 − a1 b2 (b2 + c3 )] − b1 c1 c32 + a1 b1 c3 (b2 + c3 ) M(a2 b2 + c1 c3 ) 2 ∗ ¯ b (a − a2 )(b3 − b3 ) = 2 2 . M(a2 b2 + c1 c3 ) =
(2)
From λ13 > 0 we obtain a2 < a∗2 , which implies a1 b2 + a1 c3 − a2 b2 − c1 c3 > 0. Since v∗ > 0, we have M > 0. (ii) Define the linear function L : R3+ → R3 by L (u, v, w) = (1 − c1 u − b1 v − b2 w, −1 + a1 u − c2 v, −1 + a2 u − b3 v − c3 w). Since system (2.1) admits no positive equilibrium, the convex set S = L (intR3+ ) is disjoint from O(0, 0, 0). The convex theorem for linear operators [10] shows that there is a hyperplane π through O, which is disjoint from the convex set S. Then there is a vector h = (h1 , h2 , h3 ) = 0 orthogonal to π such that h · z < 0 for all z ∈ S. Let (u(t ), v(t ), w(t )) ∈ intR3+ be a solution of (2.1) and denote
V (u, v, w) = log (uh1 vh2 wh3 ) then we have
h2 dv h3 dw h1 du dV + + = h · z < 0, z = L (u, v, w). | = dt (2.1) u dt v dt w dt Thus V(t) is decreasing along each orbit, and no point in intR3+ may belong to the ω-limit set of the orbit by the Lyapunov Theorem [10]. Since all solutions of (2.1) are bounded by Proposition 2.1, they will not blow up to infinity, which implies that they converge to the boundary of R3+ . When condition (4.1) is not satisfied, the following Theorems 4.3 and 4.5 exhibit dynamics of (2.1), which implies extinction of species. Theorem 4.3. Let (u(t ), v(t ), w(t )) ∈ intR3+ be a solution of (2.1). (2)
(i) If λ1
(3)
(ii) If λ1
≤ 0, then limt→+∞ v(t ) = 0. ≤ 0, then limt→+∞ w(t ) = 0.
Proof. Proposition 4.2(ii) is used to prove the result in (i), while a similar proof can be given for (ii). We claim that there is no positive equilibrium of system (2.1). Indeed, suppose that P∗ is a positive equilibrium of (2.1). From (2) the righthand side of the first equation of (2.1), we have c1 u∗ ≤ 1. Since λ1 ≤ 0, we obtain a1 ≤ c1 . Thus a1 u∗ ≤ c1 u∗ ≤ 1. From the righthand side of the second equation of (2.1), we have
−1 + a1 u∗ − c2 v∗ ≤ −c2 v∗ < 0 which forms a contradiction that P∗ is a positive equilibrium. Therefore, it follows from Proposition 4.2 that, all positive solutions converge to the boundary of R3+ . Since O is globally asymptotically stable on the (v, w)-plane and P1 is globally asymptotically stable on the (u, v)-plane, we obtain limt→+∞ v(t ) = 0. From Theorem 4.3 we obtain the following corollary. (2)
Corollary 4.4. If λ1
(3)
≤ 0 and λ1 (2)
Theorem 4.5. Assume λ1 (3)
(2)
(3)
(2)
(3)
(2)
≤ 0, then P1 is globally asymptotically stable. (3)
> 0 and λ1
> 0.
(i) If λ12 < 0 and λ13 > 0, then P12 is globally asymptotically stable.
(ii) If λ12 > 0 and λ13 < 0, then P13 is globally asymptotically stable.
(iii) If λ12 < 0 and λ13 < 0, both P12 and P13 are asymptotically stable. A two-dimensional manifold S2 , which passes through the positive u-axis and positive equilibrium P∗ , divides intR3+ into two regions: one is the basin of attraction of P12 while the other is that of P13 as shown in Fig. 1. Proof. We prove the results by using Proposition 4.2(ii) where it is assumed that there is no positive equilibrium of (2.1). Suppose that system (2.1) admits a positive equilibrium P ∗ (u∗ , v∗ , w∗ ). From (3.2), we have
v∗ M = a2 b2 + c1 c3 − a1 (b2 + c3 ) = −(a2 b2 + c1 c3 )( − 1 + a1 uˆ) a2 b2 + c1 c3 (2) λ13 r2 w∗ M = b3 (a1 − c1 ) − c2 (a2 − c1 ) − b1 (a2 − a1 ) =−
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Bi−stability of equilibria P
12
and P
13
4.5 P13 4
S2
3.5 3
w
2.5 2 1.5 1 0.5 P
12
0 0
5
10 u
15
20
0
15
10
5
20
25
30
v
Fig. 1. Bi-stability of equilibria P12 and P13 . Let a1 = 0.2, a2 = 0.4, b1 = b2 = 0.1, b3 = 0.6, r1 = 1, r2 = r3 = 0.1, ci = 0.1, i = 1, 2, 3. Then we have a2 > a∗2 = 0.3 and b3 > L(a2 ) = 2a2 − 0.3 = 0.5, which implies that both P12 and P13 are asymptotically stable. The two-dimensional manifold S2 , which passes through the positive u-axis, divides intR3+ into two regions: the region below S2 is the basin of attraction of P12 while the region above S2 is that of P13 .
¯ = −(a1 b1 + c1 c2 )( − 1 + a2 u¯ − b3 v) a1 b1 + c1 c2 (3) =− λ12 . r3 (3)
(2)
(i and ii) We give the proof for (i), while a similar one can be given for (ii). Since λ12 < 0 and λ13 > 0, we have v∗ w∗ < 0. Thus there is no positive equilibrium of (2.1). By Proposition 4.2, all positive solutions converge to the boundary of R3+ . Notice that O (resp. P13 ) is globally asymptotically stable on the (v, w)-plane (resp. (u, w)-plane), while P12 is globally asymptotically stable on the (u, v)-plane. Since equilibria O, P1 and P13 are saddle points on R3+ , no positive solution converges to O, P1 or P13 , and all positive solutions converge to P12 , which implies that P12 is globally asymptotically stable in intR3+ . (3)
(iii) Since λ12 < 0 and P12 is globally asymptotically stable on the (u, v)-plane, P12 is asymptotically stable. Similarly, from
(2)
λ13 < 0 we obtain that P13 is asymptotically stable. Let S1 and S2 be the boundaries of attraction basin of P12 and P13 , respectively. We can see that both S1 and S2 include the positive u-axis. On the other hand, since orbits on S1 are bounded and equilibria O, P1 on S1 are saddle points, there exists at least one positive equilibrium on S1 . From the righthand side of (2.1), there is at most one positive equilibrium, which implies that there exists a unique positive equilibrium P∗ ∈ S1 . Similarly, P∗ ∈ S2 . Suppose S1 = S2 . Then the surfaces S1 and S2 surround an open set in intR3+ . Since O and P1 on Si are saddle points, system ¯ by the Acyclicity Theorem, which implies that there is a positive equilibrium in . This is (2.1) is uniformly persistent on
impossible since system (2.1) admits at most one positive equilibrium P ∗ ∈ S1 S2 . Therefore, S1 = S2 = S2 , which implies that 2 3 the two dimensional manifold S divides intR+ into two regions: one is the basin of attraction of P12 while the other is that of P13 . 5. Mutualism In this section we present properties of the parasitism-predation system (2.1) based on Theorems 4.1–4.5. We discuss the effects of parameters a2 and b3 , but similar results can be given for other parameters. Recall that b3 denotes the deterrence effect
S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
673
Bifurcation on the (a ,b )−plane 2
3
b
3
IV: (u,v)/(u,w)
b =L(a ) 3
2
III: (u,w)
I: (u,v)
II: (u,v,w)
0
c
1
a02
a*2
a
2
Fig. 2. Bifurcation diagram on the (a2 , b3 )-plane. Let a1 = 0.2, b1 = b2 = 0.1, r1 = 1, r2 = r3 = 0.1, ci = 0.1, i = 1, 2, 3. Then we have = 0.15, a∗2 = 0.3 and b3 = L(a2 ) = 2a2 − 0.3. The lines a2 = a∗2 and b3 = L(a2 ) divide the first quadrant into four regions. In region I with a2 < a∗2 and b3 > L(a2 ), the label (u, v) means that cats always go extinct. In region II with a02 < a2 < a∗2 and b3 < L(a2 ), the label (u, v, w) means that the three species always coexist. In region III with a2 > a∗2 and b3 < L(a2 ), the label (u, w) means that cuckoos always go extinct. In region IV with a2 > a∗2 and b3 > L(a2 ), the label (u, v)/(u, w) means that either cuckoos or cats go extinct, where the species with higher initial density persists. a02
of cuckoos on cats, and a2 represents the efficiency of cats in converting the predation into fitness, which is determined by cats’ predation ability and the converting coefficient. In the following discussions, we assume ai > c1 , i = 1, 2, which implies that either cuckoos or cats can coexist with crows in the absence of the other by Theorem 4.3. Denote
a02 =
1 −1 + a2 u¯ , L(a2 ) = v¯ u¯
(3)
then λ12 > 0 can be written as b3 < L(a2 ), while the point (a02 , 0) is the intersection of the line b3 = L(a2 ) and the a2 -axis. Since
a02 − c1 =
b1 (a1 − c1 ) >0 b1 + c2
a∗2 − a02 =
(a1 − c1 )(b1 c3 + b2 c2 + c2 c3 ) >0 b2 (b1 + c2 )
we obtain a∗2 > a02 > c1 . As shown in Fig. 2, the lines a2 = a∗2 and b3 = L(a2 ) divide the first quadrant of the (a2 , b3 )-plane into four regions: Region I: a2 < a∗2 , b3 > L(a2 ); Region II: a02 < a2 < a∗2 , b3 < L(a2 ); Region III: a2 > a∗2 , b3 < L(a2 ); Region IV: a2 > a∗2 , b3 > L(a2 ). In region I, cats go extinct by Theorems 4.3 and 4.5(i). In region II, all species persist by Theorem 4.1. In region III, cuckoos go extinct by Theorem 4.5(ii). In region IV, there exists bi-stability of equilibria P12 and P13 by Theorem 4.5(iii). Parameter a2 varies with the cats’ predation ability b2 and efficiency e, and a large a2 ( > a∗2 ) implies an intensive predation b2 or a high efficiency e. Indeed, we have a2 = er1 b2 /r3 by Section 2. Thus, a2 > a02 can be written as e > e0 = r3 a02 /(r1 b2 ), or
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b2 > b02 = r3 a02 /(er1 ), while a2 > a∗2 can be written as e > e∗ = r3 a∗2 /(r1 b2 ). From a2 > a∗2 and (4.2), we obtain
er1 b22 − r3 a1 b2 − c3 r3 (a1 − c1 ) > 0 which implies that a2 > a∗2 can be written as b2 > b∗2 with
b∗2
=
r3 a1 +
r32 a21 + 4c3 r3 (a1 − c1 ) 2er1
.
Therefore, a large a2 ( > a∗2 ) implies an intensive predation b2 ( > b∗2 ) or a high efficiency e( > e∗ ). Theorems 4.1 –4.5 demonstrate basic properties of the crow–cuckoo–cat system. We now offer a biological interpretation of the mathematical results presented in Section 4. First, the three species can coexist when a2 is intermediate and b3 is small. As shown in region II of Fig. 2 and Theorem 4.1, system (2.1) is persistent if a02 < a2 < a∗2 and b3 < L(a2 ). The biological reason is that a weak deterrence (b3 ) will not result in extinction of cats and an intermediate predation/efficiency (a2 ) will not lead to extinction of cuckoos, which allows for the coexistence of the three species. ˆ where u, ¯ uˆ and u∗ are defined Second, when coexisting at an equilibrium, crows and cuckoos are mutualistic only if u¯ > u, ∗ ¯ Thus, if u¯ ≤ u, ˆ we obtain u∗ − uˆ < 0, which in (2.4), (2.6) and (3.2), respectively. As shown in Proposition 3.3, we have u < u. implies that interaction outcomes between crows and cuckoos are not mutualistic. The reason is that u¯ ≤ uˆ means the negative effect of cuckoos on crows is greater than or equal to that of cats. Thus, although cuckoos can decrease the predation of cats, they have a larger negative effect on the growth of crows, which cannot lead to mutualism between them. ˆ interaction outcomes between crows and cuckoos can change from Third, when coexisting at an equilibrium and u¯ > u, parasitism to commensalism and mutualism as b3 increases. To show this change, we need to display that when the positive equilibrium P ∗ (u∗ , v∗ , w∗ ) is asymptotically stable, the sign of u∗ − uˆ changes from negative to positive, where uˆ comes from ˆ 0, w ˆ ). Because of the complexity of u∗ and conditions in Theorem 3.2, we consider a specific case. Fix the equilibrium P13 (u, a1 = 0.2, a2 = 0.25, b1 = b2 = 0.1, r1 = 1, r2 = r3 = 0.1, ci = 0.1 for i = 1, 2, 3 and let b3 < L(a2 ) = 0.2 vary. A direct computation shows that
a02 = 0.15, a∗2 = 0.3, u¯ = u∗ =
10(0.3 − b3 ) , 0.55 − b3
v∗ =
20 10 , v¯ = , L(a2 ) = 2a2 − 0.3 3 3
5 10(4.7 − b3 ) 40 . , w∗ = , uˆ = 0.55 − b3 0.55 − b3 7
Since b3 < 0.2, P ∗ (u∗ , v∗ , w∗ ) is a positive equilibrium. Moreover, since
4a2 b1 c2 c3 = 0.001 > a1 b2 b23 = 0.02b23
as
b3 < 0.2
P∗ is globally asymptotically stable by Theorem 3.4. It follows from
u∗ − uˆ =
100(b3 − 0.1) 7(0.55 − 2b3 )
that interaction outcomes between crows and cuckoos change from parasitism to commensalism and mutualism as b3 increases from b3 < 0.1 to b3 = 0.1 and b3 > 0.1. Numerical simulations in Fig. 3 confirm the change when b3 = 0.05, b3 = 0.1 and b3 = 0.15. The biological reason for the change from parasitism to commensalism and mutualism is as follows. In this parameter region, the predation/efficiency of cats is intermediate. When the deterrence b3 is weak, cuckoos do little contribution to decrease the loss of crows by cats. This implies that the relationship is parasitic since cuckoos depend on crows for persistence. When the deterrence b3 is strong, the crow and cuckoo are mutualistic since a strong deterrence b3 decreases the predation, which decreases the loss of crows by cats compared with that by cuckoos. Fourth, there exists extinction of species when the values of a2 and b3 are unbalanced, i.e., the condition of a02 < a2 < a∗2 , b3 < L(a2 ) is not satisfied. As shown in region I of Fig. 2 and Theorem 4.5(i), when a02 < a2 < a∗2 but b3 > L(a2 ), cats go extinct. The reason is that an extremely strong deterrence b3 prohibits the predation and results in the extinction of cats. Moreover, when c1 < a2 < a02 and b3 > 0, cats go extinct. The reason is that the cats have a very weak predation/efficiency (a2 ) that cannot endure any deterrence (b3 ) from cuckoos. Since cats can survive in the crow-cat system, it is the cuckoo that leads to the extinction. Cuckoos can also be driven into extinction by cats. As shown in region III of Fig. 2 and Theorem 4.5(ii), when a2 > a∗2 but b3 < L(a2 ), cuckoos go extinct. The reason is that a strong predation/efficiency a2 leads to explosive growth of cats and results in the extinction of cuckoos. Since cuckoos can survive in the crow-cuckoo system, it is the cats’ predation that leads to the extinction of cuckoos. Thus, a strong deterrence by a cuckoo not only benefits its host, but also provides itself protection from extinction. Finally, initial densities play a role in survival of species. As shown in region IV of Fig. 2 and Theorem 4.5(iii), P12 and P13 are asymptotically stable when a2 > a∗2 and b3 > L(a2 ). Thus, initial densities in the attraction region of P12 means survival of cuckoos, while those in the attraction region of P13 means survival of cats. The reason is that in the situation where the predation/efficiency is strong and the deterrence is effective, the species with large initial density will be dominant. Since cuckoos and cats can coexist with crows individually, it is the initial density that determines their survival in the three-species system.
S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
(a) Parasitism at b =0.01
(b) Commensalism at b =0.1
3
3
population density
10
10 crow cuckoo cat
8
6
4
4
2
2
0
10
20
30
crow cuckoo cat
8
6
0
40
0
0
population density
(c) Mutualism at b3=0.19 10
8
8
6
6
4
4
2
2
0
50
100 t−time
150
20
40
60
(d) Mutualism at b3=0.8
10
0
675
200
0
0
10
20 t−time
30
40
Fig. 3. Transitions of interaction outcomes between crows and cuckoos from parasitism to commensalism and mutualism when b3 increases. Fix a1 = 0.2, a2 = 0.25, b1 = b2 = 0.1, r1 = 1, r2 = r3 = 0.1, ci = 0.1, i = 1, 2, 3, and let b3 vary. Then we have L(a2 ) = 0.2, which implies that the three species coexist when b3 < 0.2. Here, the density of crows in the crow–cat subsystem is uˆ = 5.7143, which is shown in the red dashed line. Solutions of u, v and w are represented by red, blue and black curves, respectively. (a) When b3 = 0.01 < 0.1, the outcomes are parasitism. (b) When b3 = 0.1, the outcomes are commensalism. (c) When b3 = 0.19 > 0.1, the outcomes are mutualism. (d) When b3 = 0.8 > 0.2, cats are driven into extinction and the outcomes are mutualism. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
6. Discussion In this paper, we consider mutualism in a parasitism-predation system consisting of crows, cuckoos and cats. Dynamics of the three-species model demonstrate the mechanism by which the crow and cuckoo can be mutualistic in the presence of cats, as shown in Theorems 4.1–4.5. Results in this work not only explain biological observation by Canestrari et al. [4], but also predict situations that have not been observed by Canestrari et al. [4]. We focus on parameter a2 , which represents both the predation degree and efficiency of cats in converting the predation into fitness, as shown in Section 2. For simplicity, we assume u¯ > uˆ and the system persists at a steady state P ∗ (u∗ , v∗ , w∗ ). Recall that u¯ and uˆ denote crows’ density in the crow–cuckoo subsystem and crow-cat subsystem, respectively. (A) When a2 is small (i.e., c1 < a2 < a02 ), Theorem 4.5(i) exhibits that cats will be driven into extinction by cuckoos, which implies that crows approach a high density u¯ ( > uˆ). Thus, the crow and cuckoo are mutualistic in this case. (B) Assume that a2 is intermediate (i.e., a02 < a2 < a∗2 ). If the negative effect b3 of cuckoos on cats is small (b3 < L(a2 )), Theorem 4.1 displays that the three species coexist and crows will approach a high density u∗ ( > uˆ) by (4.3). Thus, the crow and cuckoo are mutualistic, which explains the observation by Canestrari et al. [4]. If b3 > L(a2 ), Theorem 4.5(i) exhibits that cats will be driven into extinction by cuckoos, which implies that the crow and cuckoo are mutualistic as shown in (A). (C) Assume that a2 is large (i.e., a2 > a∗2 ). If b3 < L(a2 ), Theorem 4.5(ii) displays that cuckoos will be driven into extinction by cats, which implies that crows approach the ˆ Thus, the crow and cuckoo are neutralistic. If b3 > L(a2 ), Theorem 4.5(iii) displays that either cuckoos or cats go extinct. density u. ˆ When cats go extinct, the When cuckoos go extinct, the crow and cuckoo are neutralistic since crows approach the density u. crow and cuckoo are mutualistic since crows approach a high density u¯ ( > uˆ). Periodic oscillations can occur in the crow–cuckoo–cat system. As shown in Theorem 3.2(ii), there exists a periodic solution of system (2.1) when a Hopf bifurcation appears. For example, let r1 = r2 = r3 = 1, a1 = 0.4, a2 = 0.7, b1 = b2 = 0.1, b3 = 0.00001, c1 = 0.01, c2 = 0.1, c3 = 0.00001. Then the solution of system (2.1) with initial value (0.3,0.3,0.3) is in periodic oscillation, as shown in Fig. 4d.
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S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
(a) b =0.5 with initial value (0.3,0.8,0.1)
(b) b =5.5 with initial value (0.3,0.8,0.1)
3
population density
10
3
10 crow cuckoo cat
8 6
6
4
4
2
2
0
0
10
20
30
crow cuckoo cat
8
40
0
0
population density
(c) b3=5.5 with initial value (0.3,0.1,0.8)
10
20
30
40
(d) Hopf bifurcation in system (2.1)
10
60
8
50 40
6 30 4 20 2 0
10 0
10
20 t−time
30
40
0
0
200
400 t−time
600
800
Fig. 4. Extinction of species and Hopf bifurcation. In (a)–(c), fix a1 = 0.2, a2 = 0.4, b1 = b2 = 0.1, r1 = 1, r2 = r3 = 0.1, ci = 0.1 for i = 1, 2, 3, and let b3 vary. Then we have a2 > a∗2 = 0.3, which implies that P13 is asymptotically stable. Here, L(a2 ) = 0.5. Solutions of u, v and w are represented by red, blue and black curves, respectively. (a) When b3 = 0.5, cuckoos go extinct. (b) When b3 = 5.5 with initial value (0.3,0.8,0.1), cats are driven into extinction by cuckoos. (c) When b3 = 5.5 with initial value (0.3,0.1,0.8), cuckoos are driven into extinction by cats. (d) Hopf bifurcation. Let r1 = r2 = r3 = 1, a1 = 0.4, a2 = 0.7, b1 = b2 = 0.1, b3 = 0.00001, c1 = 0.01, c2 = 0.1, c3 = 0.00001. The solution with initial value (0.3,0.3,0.3) is periodic. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The biological observations by Canestrari et al. [4] support the crow–cuckoo–cat model in this work. From the observations by Canestrari et al. [4], we obtain the approximated parameter values in model (2.1) on MatLab by least square method of linear regression: r1 = 6.5459, r2 = 0.0816, r3 = 0.0001, r2 a1 = 0.08 r1 b2 = 0.1, r1 b3 = 0.0238 r3 a2 = 0.05, r1 b1 = 0.1, r1 c1 = 0.5919, r2 c2 = 0.1562, r3 c3 = 0.0033 with initial value (3.0084, 0.5741, 47.5491). Numerical simulations in Fig. 5 show that our model fits the biological observation well. Moreover, numerical simulations in Fig. 5 show that u∗ = 3.5018 > uˆ = 3.1353. Thus the crow and cuckoo are mutualistic, which is consistent with both Theorem 4.1 and the observation by Canestrari et al. [4]. The assumption (3.6) is consistent with the positivity of P∗ , which can be shown by simulations in Fig. 3a. Here, we have r1 = 1, r2 = r3 = 0.1, a1 = 0.2, a2 = 0.25, b1 = b2 = 0.1, b3 = 0.01, ci = 0.1, i = 1, 2, 3. Numerical simulations display that there is a positive equilibrium P∗ . Moreover, a direction computation exhibits that
4a2 b1 c2 c3 = 10−2 > 2 × 10−5 = a1 b2 b23 which implies that there is no contradiction between the assumption (3.6) and the positivity of P∗ . Functional responses in the present model are linear. If more general functional responses (e.g., saturated responses) are used, there may exist more complicated dynamics such as chaos in the system, which will be studied in our further study. Acknowledgements We are grateful to the three anonymous reviewers for their careful reading, helpful comments and suggestions that really helped us to improve the presentation of the paper. This work was supported by Student Research Training Program of Zhejiang University, NSF of Guangdong (S2012010010320) and NSF of People’s Republic of China (11571382).
S. Wang, Y. Wang / Applied Mathematics and Computation 273 (2016) 664–677
677
Fitting the real data with model (2.3) 50
45
40
population density
35
30 crow cuckoo cat
25
20
15
10
5
0
0
5
10
15 t−time
20
25
30
Fig. 5. Fitting the empirical observation with model (2.1). The data observed by Canestrari et al. [4] is shown in red ∗, while solutions of u, v and w are represented by red, blue and black curves, respectively. Let r1 = 6.5459, r2 = 0.0816, r3 = 0.0001, r2 a1 = 0.08, r3 a2 = 0.05, r1 b1 = r1 b2 = 0.1, r1 b3 = 0.0238, r1 c1 = 0.5919, r2 c2 = 0.1562, r3 c3 = 0.0033 with initial value (3.0084, 0.5741, 47.5491). Numerical simulations show that our model fits the empirical observation well and the crow and cuckoo are mutualistic in the presence of cats. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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