Mutualisms in a parasitism–predation system consisting of crows, cuckoos and cats

Accepted Manuscript

Mutualisms in a parasitism-predation system consisting of crows, cuckoos and cats Yuanshi Wang PII: DOI: Reference:

S0307-904X(16)30157-3 10.1016/j.apm.2016.03.032 APM 11100

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Applied Mathematical Modelling

Received date: Revised date: Accepted date:

27 September 2015 8 March 2016 23 March 2016

Please cite this article as: Yuanshi Wang, Mutualisms in a parasitism-predation system consisting of crows, cuckoos and cats, Applied Mathematical Modelling (2016), doi: 10.1016/j.apm.2016.03.032

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Highlights • The crow-cuckoo-cat system consisting of parasitism, predation and mutualism, is modeled.

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• The crow-cuckoo interaction can change among parasitism, commensalism, mutualism and neutralism.

• When predation from cat is weak, cuckoo has a small positive effect on crows and their interaction is parasitism.

• When the predation is intermediate, the positive effect becomes large and the

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interaction is mutualism.

• Periodic oscillations of the population may occur in certain range of the pa-

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rameters.

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Mutualisms in a parasitism-predation cats Yuanshi Wang∗

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system consisting of crows, cuckoos and

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School of Mathematics and Computational Science

Sun Yat-sen University, Guangzhou 510275, P.R.China

Abstract An avian brood parasite usually places its eggs in the nest of other

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birds by stealth, which raise the unrelated chicks. Since the chicks compete food with the host offsprings, the host typically suffers serious loss of its own brood.

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However, an over 16-year study on cuckoos and their host (crows) demonstrates that the parasites may provide a benefit to the host by deterring predators (e.g.,

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quasi-feral cats). In order to better understand these changes, we propose a mathematical model to describe the crow-cuckoo-predator system. Dynamics of the model

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demonstrate mechanism by which the crow-cuckoo interaction can change among parasitism, commensalism, mutualism and neutralism in a smooth fashion. Our results not only consolidate empirical observations, but also reveal some new phe-

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nomena: the crow-cuckoo interaction will return from mutualism to parasitism when the predation is extremely strong, and periodic oscillations of the populations may occur in certain range of the parameters. Keywords Persistence; stability; bifurcation; periodic orbit; deterrence ∗

Corresponding author. E-mail address: [email protected]. Fax: 8620 8403 7978

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1

Introduction

In the crow-cuckoo interaction, a cuckoo lays eggs in the nest of a crow and the crow raises the unrelated chicks. Since there are predators on the crow nest, a repellent secretion by cuckoo chicks can deter the visitation of predators to the nest. Thus, the

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cuckoo has both positive and negative effects on the crow. Unexpected interactions between the cuckoo and crow had been observed by Canestrari et al. (2014), which can change from parasitism to mutualism as the intensity of predation increases.

The long-term study was performed in an area of North Spain. The parasite is the great spotted cuckoo Clamator glandarius, the host is carrion crows Corvus

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corone corone, and the crow-nest predators include mammals, corvids and raptor birds (Canestrari et al., 2014; Bolopo et al., 2015). For convenience, we use cats (that is, free-ranging cats that hunt year-round but could be attracted with food) to represent the predators in this work. While many cuckoos evict other eggs and chicks and are raised alone in the nest, the great spotted cuckoo in the study area

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does not evict host nest mates and so it is raised alongside with them. It is observed by Canestrari et al. (2014) that the cuckoo chicks reveal a mix of caustic and

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repulsive compounds that are known to repel mammals and birds. In a parasitised nest, fewer crows fledge when compared with non-parasitised nests, which displays

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a true cost of parasitism. However, the parasitised nest has a larger chance to escape predation. Thus, the cuckoos have a negative effect on the number of crows

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fledged, while they have a positive effect on the nest success because of predator repellence. These counterbalancing effects from cuckoos may make the crow-cuckoo interaction fluctuate between parasitism and mutualism, depending on the intensity

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of predation pressure. Long-term data collected by Canestrari et al. (2014) showed a close connection

between the presence of a parasitic chick and greater nest success. For example, among non-parasitized nests, the addition of cuckoo chicks leads to significantly enhanced success: the probability of success in unmanipulated non-parasitized nests is 0.375, while that in the manipulated parasitized nests is 0.714. Among parasitized nests, the removal of cuckoo chicks results in significantly reduced success: the 2

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probability of success in cuckoo-removed nests is 0.312, while that in cuckoo-hold nests is 0.607. Moreover, the long-term data also display that outcome of crowcuckoo interaction depends upon the intensity of predation pressure: during seasons with low rates of nest failure, cuckoos reduce reproductive success of crows. However,

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during breeding seasons with intensive nest predation, parasitized nests produced more fledglings than non-parasitized nests. Therefore, the crow-cuckoo interaction should not be strictly categorized as parasitism or mutualism. It fluctuates between these outcomes in different seasons, depending upon the predation pressure.

The outcomes of crow-cuckoo interaction are determined by positive (+), neutral

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(0), or negative (−) effects of one species on the other (Bronstein, 1994; Neuhauser and Fargione, 2004; Wang et al., 2015). Assume that crows can approach a density of x1 in the absence of cuckoos, while in the presence of cuckoos, they approach a density of x2 . Here, the density of a species represents the number of the species in a certain area. When cuckoos can depend upon crows for survival in the crow-

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cuckoo-cat system, the crow-cuckoo interaction is mutualism (+ +) if x2 − x1 > 0. The reason is that each species has a positive effect on the other. Similarly, the

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crow-cuckoo interaction is commensalism (+ 0) if x2 − x1 = 0, and is parasitism

(+ −) if x2 − x1 < 0. When cuckoos cannot depend upon crows for survival in

the crow-cuckoo-cat system, the crow-cuckoo interaction becomes neutralism (0 0)

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since neither species can benefit from the other. If the population of crows is in periodic oscillations, we apply the mean density of species (Hofbauer and Sigmund,

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1998) to approximate the density, as shown in Remark 5.1 of this work. Since the crow-cuckoo interactions are complex, it is necessary to establish a crow-cuckoo-cat

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model and demonstrate mechanism by which the crow-cuckoo interaction can change among mutualism, commensalism, parasitism and neutralism. Many interesting models have been formed to characterize one-resource and two-

consumer systems. Cantrell et al. (2004) formed a consumer-resource model in which there are one resource and two consumers that compete for the same resource. Applying persistence theory, they showed that the two consumer species can coexist upon the same limiting resource in the sense of permanence. Kang and

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Wedekin (2013) proposed an intraguild predation (IGP) model in which two predators consume the same resource while one of the predators is the prey of the other. Global dynamics of the model demonstrate that the model has multiple stable interior equilibria, which confirms phenomena obtained in empirical observations. Shu

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et al. (2015) considered an IGP system with time delay. Dynamics of the model demonstrate that varying the delay can lead to two stable periodic solutions. For more relevant papers, we refer to Verdy and Amarasekare (2010), Hsu et al. (2011, 2013), Wang et al. (2011,2016), Freeze et al. (2014), etc. As far as we know, there is little work on modeling parasitism-predation systems of three species in which the

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parasite has a positive effect on its host by deterring the predator.

In this paper, we consider the crow-cuckoo-cat system in which cuckoos depend upon crows for survival, and cats are predators of crow nests. Since a malodorous cloacal secretion by cuckoo chicks can deter the predators and then defend the crow nests, the cuckoos have both positive and negative effects on crows. We propose a

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mathematical model to describe the crow-cuckoo-predator system. In our model, functional response in the crow-cuckoo subsystem is assumed to be linear, and the

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crow-cat interaction has a Holling Type II form. Then the Holling Type II function is extended to a modified Holling Type II functional response by adding a term proportional to the cuckoo population in the denominator, representing the time lost

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to the cats because of the cuckoo deterrence. Dynamics of the model demonstrate mechanism by which the crow-cuckoo interaction can change among parasitism,

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commensalism, mutualism and neutralism in a smooth fashion: (i) When predation is weak, the deterrence by cuckoo chicks has a small positive effect on crows and

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the crow-cuckoo interaction is parasitism; (ii) When the predation is intermediate, the positive effect becomes large and the crow-cuckoo interaction is mutualism; (iii) When the predation is intensive, the deterrence by cuckoo chicks becomes less effective and the crow-cuckoo interaction returns to parasitism; (iv) When the predation is extremely intensive, cuckoos are driven into extinction by predators and the crowcuckoo interaction is neutralism. These changes also occur when the strength of deterrence varies. Our results consolidate empirical observations and experimental

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findings by Canestrari et al. (2014) and reveal some new phenomena not reported by Canestrari et al. (2014). That is, the crow-cuckoo interaction will return from mutualism to parasitism when the predation is extremely strong, and periodic oscillations of the population may occur in certain range of the parameters. Further

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analysis and simulations exhibit that there are multiple stable equilibria under certain conditions and when the equilibria disappear, the three species can coexist in periodic oscillations in the absence of interior equilibrium. Numerical computations show that our model fits the empirical observations well.

The paper is organized as follows. The crow-cuckoo-cat model is characterized

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in Section 2. Section 3 exhibits persistence of the system and Section 4 shows bifurcation analysis, both of which theoretically prove all possible outcomes of crowcuckoo interactions. Section 5 displays transition of crow-cuckoo interactions by numerical simulations. Discussions are in Section 6.

Model

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2

In this section, we describe the crow-cuckoo-cat system by a mathematical model,

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show boundedness of solutions of the model, and exhibit dynamics of the subsystems. In the crow-cuckoo subsystem, cuckoos depend upon crows for survival and then

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crow-cuckoo interaction is parasitism. For simplicity, we use the Lotka-Volterra par-

x + β1 y dx = r1 x(1 − ), dt K dy = y(−r2 + α1 x − d2 y), dt

(2.1)

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asitism model with a Holling Type I functional response to describe the interaction:

where x and y represent population densities of crows and cuckoos, respectively. The biological meaning of model (2.1) is explained as follows. Let y = 0 in the first equation of (2.1), we obtain dx/dt = r1 x(1 − x/K). Thus, parameter r1 represents

the intrinsic growth rate of crows, and K is the carrying capacity when in isolation. Let x = 0 in the second equation of (2.1), we obtain dy/dt = y(−r2 − d2 y). Thus, parameter r2 represents the death rate of cuckoos and d2 is the self-incompatible degree. Moreover, the term “−β1 y” in model (2.1) represents the parasitism of 5

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cuckoos and parameter β1 denotes the quantity of resources consumed by a cuckoo, while α1 represents its efficiency in converting the consumption into fitness. In the crow-cat subsystem, crow-cat interaction is predation in which cats are predators. Thus, we use the Lotka-Volterra predator-prey model with a Holling

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Type II functional response to describe the interaction: x β2 xz dx = r1 x(1 − ) − , dt K c1 + x (2.2) dz eβ2 x = z(−r3 + ), dt c1 + x where x and z represent population densities of crows and cats, respectively. Pa-

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rameters r1 and K have the same meaning as those in (2.1), while r3 represents the death rate of cats. Moreover, the term “−β2 xz/(c1 + x)” in model (2.2) represents the predation by cats. Since limx→∞ β2 x/(c1 + x) = β2 , parameter β2 represents the saturation level in predation. When x = c1 , we obtain β2 x/(c1 + x) = β2 /2. Thus c1 is the half-saturation density of crows and we have c1 < K. Moreover, e represents

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the cats’ efficiency in converting the predation into fitness.

In a crow-cuckoo-cat system, a cuckoo can interfere with a cat by deterring its

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predation on the crow nest by derivation of a mix of caustic and repulsive compounds. Thus the cuckoo acts as an interferer to the cat, which leads to a loss in time to the cat. Then the Holling Type II function describing crow-cat interaction

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can be reasonably extended to a modified Holling Type II functional response by adding a term proportional to y in the denominator, representing the time lost to the

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cats. On the other hand, experimental findings show that cuckoos with the caustic and repulsive compounds, will not be attacked by cats (Page 1351, Canestrari et al.,

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2014; Ruer et al. 2014). Thus, the crow-cuckoo-cat system can be depicted by dx β2 z = r1 x(1 − d1 x − β1 y − ), dt c1 + x + c2 y dy = r2 y(−1 + α1 x − d2 y), dt dz α2 x = r3 z(−1 + ), dt c1 + x + c2 y

where we use the replacement: d1 :=

1 β1 β2 α1 d2 eβ2 , β1 := , β2 := , α1 := , d2 := , α2 := . K K r1 r2 r2 r3 6

(2.3)

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Parameter c2 represents the lost time of cats due to deterrence by a cuckoo. From c1 < K we have 1 − c1 d1 > 0. All parameters in (2.1) and (2.2) are positive and their biological meanings are displayed in Table 1.

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Table 1. ———————————————————————————————– Parameter

Biological meaning

———————————————————————————————————— r1

The intrinsic growth rate of crows

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———————————————————————————————————— r2

The death rate of cuckoos

———————————————————————————————————— r3

The death rate of predators

———————————————————————————————————— The carrying capacity of crows

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K

———————————————————————————————————— Cuckoos’ efficiency in converting their consumption into fitness

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α1

———————————————————————————————————— β1

The quantity of resources consumed by a cuckoo

β2

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———————————————————————————————————— Predators’ saturation level in predation

e

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———————————————————————————————————— Predators’ efficiency in converting their predation into fitness

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———————————————————————————————————— c1

The half-saturation density in predation

———————————————————————————————————— c2

The lost time of predators due to deterrence by a cuckoo

———————————————————————————————————— d2

The self-incompatible degree in cuckoos

———————————————————————————————–

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We consider solutions of system (2.3) under the initial value conditions x(0) ≥ 0, y(0) ≥ 0, z(0) ≥ 0.

Dissipativity

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2.1

(2.4)

First of all, we can see that the solutions to the initial value problem (2.3)-(2.4) are nonnegative. We also have the following results on the boundedness of solutions of system (2.3).

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Proposition 2.1. System (2.3) is dissipative.

Proof. By the first equation in system (2.3), we obtain dx ≤ r1 x(1 − d1 x). dt

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Thus, the comparison principle (Cosner, 1996) implies that lim sup x(t) ≤

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t→∞

1 , d1

so that for  > 0 small, we obtain x(t) ≤  + 1/d1 when t is sufficiently large. Let

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r0 = min{r2 , r3 }. From the three equations in (2.3), we obtain

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d r1 β1 r1 β2 r1 β1 d2 r1 β1 2 r1 β2 (x + y+ z) = r1 x(1 − d1 x) − r2 y− y − r3 z dt r2 α1 r3 α2 r2 α 1 α1 r3 α2 r1 β1 r1 β2 < (r1 + r0 )x − r0 (x + y+ z) r2 α 1 r3 α2 1 r1 β1 r1 β2 ≤ (r1 + r0 )( + ) − r0 (x + y+ z). d1 r2 α1 r3 α2

Citing the comparison theorem again, we obtain lim sup(x + t→∞

r1 β1 r1 β2 1 1 y+ z) ≤ (r1 + r0 )( + ), r2 α 1 r3 α 2 r0 d1

which means that system (2.3) is dissipative.

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2.2

Subsystems

There are three two-species subsystems. In the cuckoo-cat subsystem (i.e., the (y, z)-subsystem), we can see that all solutions converge to the original point, which implies that both cuckoos and cats go extinct. The reason is that cuckoos (y) and

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cats (z) are assumed to depend upon crows (x) for survival. In the following, we cite some known results on the crow-cuckoo and crow-cat subsystems (i.e., the (x, y)and (x, z)-subsystems). 2.2.1

Subsystem I: Holling Type I

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We first consider the crow-cuckoo subsystem with Holling Type I functional response:

(2.5)

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dx = r1 x(1 − d1 x − β1 y), dt dy = r2 y(−1 + α1 x − d2 y). dt

The subsystem (2.5) has been well studied. It has two equilibria on the axes, namely O(0, 0) and P1 (1/d1 , 0). O is a saddle point with eigenvalues r1 and −r2 , while P1 α1 > d1 , where

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has eigenvalues −r1 , r2 (−1 + α1 /d1 ). (2.5) has an interior equilibrium P12 (x+ , y + ) if

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x+ =

β1 + d2 α 1 − d1 , y+ = . α1 β1 + d1 d2 α 1 β1 + d1 d2

(2.6)

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Here, we summarize dynamics of (2.5) by Hofbauer and Sigmund (1998) as follows. Proposition 2.2. If α1 > d1 , then equilibrium P12 is globally asymptotically stable.

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Otherwise, solutions of (2.5) with positive initial values converge to P1 . Proposition 2.2 exhibits mechanism by which cuckoos can depend upon crows

for survive. When cuckoos’ efficiency in converting their consumption in crow nests into fitness is large (α1 > d1 ), cuckoos and crows coexist at a steady state (P12 ). Otherwise, cuckoos go extinct and crows approach the carrying capacity (1/d1 ). The reason of extinction is that cuckoos’ efficiency is small.

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2.2.2

Subsystem II: Holling Type II

Now we consider the crow-cat subsystem with Holling Type II functional response:

(2.7)

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dx β2 z = r1 x(1 − d1 x − ), dt c1 + x α2 x dz = r3 z(−1 + ). dt c1 + x

The subsystem (2.7) has been well studied. It has two equilibria on the axes, namely ¯ 0) and P¯1 (1/d1 , 0). O ¯ is a saddle point with eigenvalues r1 and −r3 , while P¯1 has O(0, eigenvalues −r1 , r3 [−1 + α2 /(1 + c1 d1 )]. (2.7) has an interior equilibrium P13 (¯ x+ , z + ) x¯+ =

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if α2 > 1 + c1 d1 , where

c1 1 , z + = (1 − d1 x¯+ )(c1 + x¯+ ). α2 − 1 β2

(2.8)

Here, we summarize dynamics of (2.7) by Kuang and Freedman (1988) as follows. Proposition 2.3.

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(i) If α2 ≤ 1 + c1 d1 , then P¯1 is globally asymptotically stable.

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(ii) If 1 + c1 d1 < α2 ≤ 1 + 2c1 d1 /(1 − c1 d1 ), then P13 is globally asymptotically stable.

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(iii) If α2 > 1 + 2c1 d1 /(1 − c1 d1 ), then P13 is unstable and there is a unique limit cycle φ surrounding P13 . All solutions of (2.7) with positive initial values

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(except P13 ) converge to φ. Proposition 2.3 demonstrates mechanism by which cats can depend upon crows

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for survive. (i) When cats’ efficiency in converting their predation into fitness is small, cats go extinct and crows approach the carrying capacity. (ii) When the efficiency is intermediate, cats and crows coexist at a steady state. (iii) When the efficiency is large, cats and crows coexist in periodic oscillations. The reason of periodic oscillations is that a high density of crows will lead to a high density of cats, which results in large decrease of crows because of cats’ high efficiency. Then the decrease of crows results in decrease of cats, which leads to increase of crows in return. 10

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3

Persistence

In this section, we consider persistence of system (2.3). Thus we need to analyze 3 the dynamics on the boundaries in the positive octant intR+ .

Let f (x, y, z), g(x, y, z), h(x, y, z) be the functions on the righthand side of

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system (2.3), respectively. Then the Jacobian matrix of system (2.3) is    fx fy fz     J(x, y, z) =   gx gy 0  ,   hx hy hz fx = r1 (1 − d1 x − β1 y − fy = r1 x[−β1 +

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where

(3.1)

β2 z β2 z ) + r1 x[−d1 + ], c1 + x + c2 y (c1 + x + c2 y)2

c2 β 2 z r1 β2 x ], fz = − , gx = r2 α1 y, 2 (c1 + x + c2 y) c1 + x + c2 y

r3 α2 z(c1 + c2 y) , (c1 + x + c2 y)2 r3 c2 α2 xz α2 x hy = − , hz = r3 (−1 + ). 2 (c1 + x + c2 y) c1 + x + c2 y We now exhibit the equilibria and periodic solutions on the boundaries.

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gy = r2 (−1 + α1 x − 2d2 y), hx =

(a) O = (0, 0, 0). The trivial equilibrium O always exists and is a saddle point

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with eigenvalues r1 , −r2 and −r3 . Thus the (y, z)-plane is the stable subspace and the x-axis is the unstable subspace.

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(b) P1 = (1/d1 , 0, 0). The semi-trivial equilibrium P1 always exists and has

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eigenvalues

(1)

(1)

(2)

λ1 = −r1 , λ1 = r2 (−1 + (2)

Here, λ1 , λ1

(3)

and λ1

α1 α2 (3) ), λ1 = r3 (−1 + ). d1 1 + c1 d 1

(3.2)

denote the eigenvalues of equilibrium P1 in the x, y, z-

directions, respectively. (c) P12 = (x+ , y + , 0). P12 is a boundary equilibrium on the (x, y)-plane, where x+ and y + are given in (2.6). The eigenvalue of P12 in the z-direction is (3)

λ12 = r3 (−1 +

α2 x+ ). c1 + x + + c2 y + 11

(3.3)

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(d) P13 = (¯ x+ , 0, z + ). P13 is a boundary equilibrium on the (x, z)-plane, where x¯+ and z + are given in (2.8). The eigenvalue of P13 in the y-direction is (2)

λ13 = r2 (−1 + α1 x¯+ ).

(3.4)

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(e) Pφ = (φ1 , 0, φ3 ). If the condition in Proposition 2.3(iii) is satisfied, then the equilibrium P13 on the (x, z)-plane is unstable and there is a unique stable limit cycle on the (x, z)-plane, denoted by (φ1 , φ3 ). Thus (φ1 , 0, φ3 ) is a boundary periodic solution for the whole system (2.3). Because Pφ is stable on the (x, z)-plane, we only need to analyze its stability in the y-direction. Let T be the period of the periodic

Thus, if (2) λφ

r2 = T

Z

T

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solution Pφ . Then the Floquet multiplier in the y-direction is Z r2 T (−1 + α1 φ1 (t))dt]. exp[ T 0 (−1 + α1 φ1 (t))dt > 0,

0

(3.5)

(2)

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then Pφ is unstable in the y-direction. If λφ < 0, it is locally asymptotically stable.

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To state our main result in this section, we introduce the definition of persistence and permanence as follows (Freedman and Moson, 1990).

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Definition 1. (Persistence of one species) Species x is said to be persistent in system (2.3) if there are constants 0 < m < M , such that for any initial values with

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x(0) > 0, the solution of (2.3) satisfies m ≤ lim inf x(t) ≤ lim sup x(t) ≤ M. t→∞

t→∞

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A similar definition can be given for species y and z. Definition 2. (Permanence of a system) System (2.3) is said to be permanent if there are constants 0 < m < M , such that for any initial values with x(0) > 0, y(0) > 0, z(0) > 0, the solution of (2.3) satisfies m ≤ lim inf min{x(t), y(t), z(t)} ≤ lim sup max{x(t), y(t), z(t)} ≤ M. t→∞

t→∞

A similar definition can be given for subsystems (2.5) and (2.7). 12

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3 Theorem 3.1. (Persistence of crows x) Species x is persistent in R+ for system

(2.3). Moreover, if α1 < d1 and α2 < 1 + c1 d1 , then P1 (1/d1 , 0, 0) is globally 3 asymptotically stable in intR+ .

Proof. Since the ω-limit set of the (y, z)-plane of (2.3) is equilibrium O(0, 0, 0), and (1984).

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O has eigenvalues r1 , −r2 , −r3 , species x is persistent by Theorem 2.5 of Hutson If α1 < d1 , then any solution of system (2.3) with y(0) > 0 satisfies limt→∞ y(t) = 0. Indeed, denote

By the first equation of (2.3), we have

d1 − α1 . 2d1 α1

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0 =

dx ≤ r1 x(1 − d1 x), dt

so that for 0 > 0, we have x(t) ≤ 0 + 1/d1 as t is sufficiently large. By the second

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equation of (2.3), we have

dy ≤ r2 y(−1 + α1 x) ≤ −r2 α1 0 y < 0, dt

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which implies that limt→∞ y(t) = 0.

Assume α2 ≤ 1. When t is sufficiently large, we have

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dz −c1 + (α2 − 1)x −c1 r3 ≤ r3 z ≤ z < 0, dt c1 + x c1 + 0 + 1/d1

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which implies that limt→∞ z(t) = 0.

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Assume 1 < α2 < 1 + c1 d1 . Denote e0 =

c1 1 − . α 2 − 1 d1

From α2 < 1 + c1 d1 we have e0 > 0. Since lim supt→∞ x(t) ≤ 1/d1 , we obtain x(t) < e0 /2 + 1/d1 as t is sufficiently large. Then we have dz r3 (α2 − 1)z c1 r3 e0 (α2 − 1)z r3 e0 (α2 − 1) = (x − )<− <− z < 0, dt c1 + x α2 − 1 2(c1 + x) 2(c1 + e0 /2 + 1/d1 ) which implies that limt→∞ z(t) = 0. Therefore, Theorem 3.1 is proven. 13

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Theorem 3.1 exhibits mechanism by which crows persist. Since it is assumed that both cuckoos and cats depend upon crows for survival, crows always persist and will not be driven into extinction by the invasion of cuckoos and cats. Moreover, when both cuckoos and cats have small efficiencies, they go extinct and crows approach

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the carry capacity. Theorem 3.2. (Persistence of crows x and cuckoos y) Species x and y are persistent 3 in R+ for system (2.3) if one of the following conditions is satisfied:

(i) α1 > d1 , α2 < 1 + c1 d1 .

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(ii) α1 > d1 , 1 + c1 d1 < α2 < 1 + (2c1 d1 )/(1 − c1 d1 ) and α2 < 1 + α1 c1 . (iii) α1 > d1 , 1 + (2c1 d1 )/(1 − c1 d1 ) < α2 < 1 + α1 c1 and α1 > T [

RT 0

φ1 (t)dt]−1 .

Proof. Since species x is persistent for system (2.3), we only need to focus on persistence of species y. From α1 > d1 , species y is persistent for the (x, y)-subsystem

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(2.5) by Proposition 2.2. From Proposition 2.3, there are three types of dynamics in the (x, z)-subsystem (2.7). Therefore, according to Theorem 2.5 of Hutson (1984), (2)

(3)

ED

species y is persistent for system (2.3) if one of the following conditions is satisfied: (2)

(3)

(2)

(i) λ1 > 0, λ1 < 0; (ii) λ1 > 0, λ1 > 0, α2 < 1 + 2c1 d1 /(1 − c1 d1 ) and λ13 > 0; (2)

(2)

(2)

PT

(iii) λ1 > 0, α2 > 1 + 2c1 d1 /(1 − c1 d1 ), λ13 > 0 and λφ > 0, which means that Theorem 3.2 is proven.

CE

Theorem 3.2 demonstrates mechanism by which cuckoos persist. As shown in Theorem 3.2(i), when cats’ efficiency is small but cuckoos’ efficiency is intermediate,

AC

cats go extinct, while crows and cuckoos coexist. As shown in Theorem 3.2(ii), when cats’ efficiency is intermediate and cuckoos’ efficiency is large, cuckoos can persist. Moreover, as shown in Theorem 3.2(iii), when cats’ efficiency is large and cuckoos’ efficiency is extremely large, cuckoos can still persist. Therefore, cuckoos can persist if their efficiency is relatively larger than that of cats. Theorem 3.3. (Persistence of crows x and cats z) Species x and z are persistent 3 in R+ for system (2.3) if one of the following conditions is satisfied:

14

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(i) α1 < d1 , α2 > 1 + c1 d1 . (ii) α1 > d1 , α2 > max{1 + c1 d1 , 1 +

c1 (α1 β1 + d1 d2 ) + c2 (α1 − d1 ) }. β1 + d2

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Proof. Since species x is persistent for system (2.3), we only need to focus on persistence of species z. From α2 > 1+c1 d1 , species z is persistent for the (x, z)-subsystem (2.7) by Proposition 2.3. Therefore, according to Theorem 2.5 of Hutson (1984), species z is persistent for system (2.3) if one of the following conditions is satisfied: (2)

(3)

(2)

(3)

(3)

(i) λ1 < 0, λ1 > 0; (ii) λ1 > 0, λ1 > 0 and λ12 > 0, which means that Theorem

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3.3 is proven.

Theorem 3.3 demonstrates mechanism by which cats persist. As shown in Theorem 3.3(i), when cuckoos’ efficiency is small but cats’ efficiency is large, cuckoos go extinct, while crows and cats coexist. As shown in Theorem 3.3(ii), when cuckoos’ efficiency is large but cats’ efficiency is extremely large, cats can persist. Therefore,

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cats can persist if their efficiency is relatively larger than that of cuckoos. To exhibit uniform persistence of the whole system (2.3), we use the Acyclicity

ED

Theorem of Bulter, Freedman and Waltman (1986a, 1986b) and Freedman et al., (1994). Thus we must guarantee that the boundary equilibria and periodic solution Theorem.

PT

cannot form a heteroclinic cycle, which is the acyclicity condition in the Acyclicity Since the presence of cuckoos (resp. cats) has negative effect on the growth of

CE

cats (resp. cuckoos), the crow-cuckoo-cat system can persist only if both of the (2)

(3)

crow-cuckoo and crow-cat subsystems can persist. That is, λ1 > 0, λ1 > 0. Recall (3)

AC

that O is always a saddle point. P12 is unstable in the z-direction if λ12 > 0. P13 is (2)

(2)

unstable in the y-direction if λ13 > 0. Pφ is unstable in the y-direction if λφ > 0.

Therefore, we conclude the main result in this section. (2)

(3)

Theorem 3.4. (Persistence of the crow-cuckoo-cat system) Assume λ1 > 0, λ1 > (3)

(2)

0, λ12 > 0 and λ13 > 0 as described in detail in (3.6). System (2.3) is permanent if one of the following conditions is satisfied: (i) 1 + c1 d1 < α2 < 1 + 2c1 d1 /(1 − c1 d1 ). 15

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(ii) α2 > 1 + 2c1 d1 /(1 − c1 d1 ) and α1 > T [

RT 0

φ1 (t)dt]−1 .

Theorem 3.4 demonstrates mechanism by which the crow-cuckoo-cat system per(2)

(3)

(3)

(2)

sist. The assumption λ1 > 0, λ1 > 0, λ12 > 0 and λ13 > 0 in Theorem 3.4 can be written as inequalities c1 (α1 β1 + d1 d2 ) + c2 (α1 − d1 ) }. β1 + d2 (3.6)

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α1 > max{d1 , (α2 − 1)/c1 }, α2 > max{1 + c1 d1 , 1 +

We can see that the above inequalities imply that efficiencies of cuckoos and cats should be in a balanced region. In this region, if cats’ efficiency is intermediate,

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then the three species persist as shown in Theorem 3.4(i). If cats’ efficiency is large, then the three species can persist if cuckoos’ efficiency is relatively larger, as shown in Theorem 3.4(ii). The reason is that the relatively larger efficiency can lead to more cuckoos, which can result in strong deterrence and more crows for the survival

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of cuckoos in return.

The following result shows extinction of species. Denote

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y0 =

α 1 − d1 . d1 d2

PT

Theorem 3.5. (Extinction of one species) (i) Cuckoos y goes to extinction if α1 < d1 , or d1 < α1 < α2 /(c1 + d−1 1 + c2 y0 ).

CE

(ii) Cats z goes to extinction if α2 < 1 + c1 d1 . Proof. (i) If α1 < d1 , then limt→∞ y(t) = 0 by the proof of Theorem 3.1.

AC

Assume α1 > d1 . Then y0 > 0. Since lim supt→∞ x(t) ≤ 1/d1 , then for  > 0

small, we obtain x(t) < (1 + )/d1 as t is sufficiently large. Then we have α1 (1 + ) dy ≤ r2 (−1 + − d2 y) dt d1

as t is sufficiently large. Thus we obtain lim sup y(t) ≤ t→∞

α1 (1 + ) − d1 . d1 d2

16

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Let  → 0, we have lim supt→∞ y(t) ≤ y0 . Since α1 <

c1 +

α2 , −1 d1 + c2 y0

there is 0 > 0 such that c1 +

d−1 1

α2 . + 0 + c2 (y0 + 0 )

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α1 <

For this 0 > 0, it follows from lim supt→∞ x(t) ≤ 1/d1 and lim supt→∞ y(t) ≤ y0 that x(t) < 0 + 1/d1 and y(t) < 0 + y0 , as t is sufficiently large. r

Let V (y, z) = yz

− r2 3

. Then we obtain

= r2 (α1 x − d2 y −

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r r α2 x dV − 2 − 2 |(2.3) = r2 yz r3 (−1 + α1 x − d2 y) − r2 yz r3 (−1 + ) dt c1 + x + c2 y

α2 x )V. c1 + x + c2 y

Thus, as t is sufficiently large, we have

M

dV α2 |(2.3) ≤ r2 x(α1 − )V < 0, −1 dt c1 + d1 + 0 + c2 (y0 + 0 )

ED

which implies that limt→∞ V = 0. Since z is bounded, we have limt→∞ y(t) = 0. (ii) The result is proven in Theorem 3.1.

PT

Theorem 3.5 demonstrates mechanism by which cuckoos and cats go extinct. Since the cases of α1 < d1 and α2 < 1 + c1 d1 have been discussed in Propositions 2.2

CE

and 2.3 respectively, we focus on the case of d1 < α1 < α2 /(c1 + d−1 1 + c2 y0 ). It follows from α1 > d1 and Proposition 2.2 that cuckoos can survive in the crow-cuckoo

AC

subsystem. However, when there exist cats, cuckoos go extinct. The reason is that in the presence of cats, crows decrease and cannot provide sufficient resources for the survival of cuckoos with an intermediate efficiency α1 < α2 /(c1 + d−1 1 + c2 y0 ). Since cuckoos can survive in the absence of cats, it is the invasion of cats that results in the extinction. By the eigenvalues of equilibrium P12 , P12 is locally asymptotically stable if (2) λ1

(3)

> 0 and λ12 < 0. Similarly, by the eigenvalues of equilibrium P13 and Floquet 17

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multiplier of periodic orbit φ, P13 is locally asymptotically stable if 1 + c1 d1 < (2)

α2 ≤ 1 + 2c1 d1 /(1 − c1 d1 ) and λ13 < 0. φ is locally asymptotically stable if α2 > (2)

1 + 2c1 d1 /(1 − c1 d1 ) and λφ < 0.

Therefore, we conclude the following result.

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Theorem 3.6. (Density-dependent extinction of one species)

(i) Cats z with a small density will be driven into extinction by cuckoos y if α1 > d1 , α2 < 1 +

c1 (α1 β1 + d1 d2 ) + c2 (α1 − d1 ) . β1 + d2

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(ii) Cuckoos y with a small density will be driven into extinction by cats z if one of the following conditions is satisfied:

(iia) 1 + c1 d1 < α2 ≤ 1 + 2c1 d1 /(1 − c1 d1 ) and α1 < (α2 − 1)/c1 . RT (iib) α2 > max{1 + α1 c1 , 1 + 2c1 d1 /(1 − c1 d1 )} and α1 < T [ 0 φ1 (t)dt]−1 .

M

Theorem 3.6 demonstrates mechanism by which initial population density can result in extinction of species. As shown in Theorem 3.6(i), when cuckoos’ efficiency

ED

is large but cats’ efficiency is small, the cats with a small density will be driven into extinction by cuckoos. The reason is that the repellent secretion by cuckoo

PT

chicks deters the visitation of cats to crow nest, which results in extinction of cats with the small efficiency. As shown in Theorem 3.6(iia-b), when cats’ efficiency is

CE

intermediate but cuckoos’ efficiency is small, or when cats’ efficiency is large but cuckoos’ efficiency is intermediate, the cuckoos with a small density will be driven into extinction by cats. The reason is that the invasion of cats decreases population

AC

density of crows, which cannot provide sufficient resources for survival of cuckoos with a small/intermediate efficiency.

Remark 3.7. Theoretical analysis in this section demonstrates that crow-cuckoo interaction can be parasitism, commensalism, mutualism and neutralism. First, as shown in Theorem 3.5(i), when their efficiency is not large (d1 < α1 < α2 /(c1 +

18

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d−1 1 + c2 y0 )), cuckoos will be driven into extinction by cats, which implies that cuckoos and crows will not benefit each other and their interaction is neutralism (0 0). A similar discussion can be given for Theorem 3.6(ii). Second, as shown in Theorem 3.6(i), when their density is low and efficiency is

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small (α2 < (c1 + x+ + c2 y + )/x+ ), cats will be driven into extinction by cuckoos while crows approach a density x+ . Thus, the crow-cuckoo interaction is parasitism (+ −), commensalism (+ 0), or mutualism (+ +) if x+ − x¯+ < 0, x+ − x¯+ = 0, or

x+ − x¯+ > 0.

Main results in this section about persistence of species are summa-

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rized in Table 2. Table 2.

———————————————————————————————– Conditions

Persistence of species

M

———————————————————————————————————— α1 < d1 , α2 < 1 + c1 d1

Persistence of crows alone

ED

———————————————————————————————————— α1 > d1 , α2 < 1 + c1 d1

Persistence of crows and cuckoos

———————————————————————————————————— α2 < 1 + α1 c1

PT

α1 > d1 , 1 + c1 d1 < α2 < 1 + (2c1 d1 )/(1 − c1 d1 ), Persistence of crows and cuckoos

CE

————————————————————————————————————

AC

α1 > d1 , 1 + (2c1 d1 )/(1 − c1 d1 ) < α2 < 1 + α1 c1 , RT α1 > T [ 0 φ1 (t)dt]−1 Persistence of crows and cuckoos

———————————————————————————————————— α1 < d1 , α2 > 1 + c1 d1

Persistence of crows and cats

———————————————————————————————————— α2 > max{1 + c1 d1 , 1 +

α1 > d1

c1 (α1 β1 +d1 d2 )+c2 (α1 −d1 ) }, β1 +d2

Persistence of crows and cats

————————————————————————————————————

19

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α1 > max{d1 , (α2 − 1)/c1 }, α2 > max{1 + c1 d1 , 1 + 1 + c1 d1 < α2 < 1 + 2c1 d1 /(1 − c1 d1 )

c1 (α1 β1 +d1 d2 )+c2 (α1 −d1 ) }, β1 +d2

Persistence of the three species

————————————————————————————————————

CR IP T

2 )+c2 (α1 −d1 ) }, α1 > max{d1 , (α2 − 1)/c1 }, α2 > max{1 + c1 d1 , 1 + c1 (α1 β1 +dβ11d+d 2 RT α1 > T [ 0 φ1 (t)dt]−1 , α2 > 1 + 2c1 d1 /(1 − c1 d1 ) Persistence of the three species

———————————————————————————————–

4

Bifurcation analysis

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In this section, we consider bifurcations of system (2.3) at a unique interior equilibrium and at a set of interior equilibria respectively, while the biological implications are described in Section 5.

A unique interior equilibrium

M

4.1

The dissipativity and permanence of system (2.3) imply that there is an interior

ED

equilibrium P ∗ (x∗ , y ∗ , z ∗ ) (Bulter, Freedman and Waltman, 1986a). Denote α20 = 1 +

α 1 c2 . d2

PT

When α2 6= α20 , it follows from the righthand side of (2.3) that c1 d 2 − c2 , d2 α2 − d2 − α1 c2 1 + α 1 c1 − α 2 y∗ = , d2 α2 − d2 − α1 c2 1 z ∗ = (1 − d1 x∗ − β1 y ∗ )(c1 + x∗ + c2 y ∗ ). β2

AC

CE

x∗ =

The Jacobian matrix of system (2.3) at P ∗  ∗ ∗  fx fy  ∗ ∗ A∗ =   gx gy  h∗x h∗y 20

(4.1)

is fz∗



  0  ,  0

(4.2)

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where fx∗ = r1 x∗ [

β2 z ∗ c2 β2 z ∗ ∗ ∗ − d ], f = r x [ − β1 ], 1 1 y (c1 + x∗ + c2 y ∗ )2 (c1 + x∗ + c2 y ∗ )2

r1 β2 x∗ < 0, gx∗ = r2 α1 y ∗ > 0, gy∗ = −r2 d2 y ∗ < 0, c1 + x ∗ + c2 y ∗ c2 x ∗ c1 + c2 y ∗ ∗ ∗ h∗x = r3 α2 z ∗ > 0, h = −r α z < 0. 3 2 y (c1 + x∗ + c2 y ∗ )2 (c1 + x∗ + c2 y ∗ )2 The characteristic equation of A∗ is

where a1 = a2 =

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λ3 + a1 λ2 + a2 λ + a3 = 0,

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fz∗ = −

A1 α22 + B1 α2 + C1 , α2 (d2 α2 − d2 − α1 c2 )

r1 r2 y ∗ (A2 α2 + B2 ) r1 r3 (α2 − 1)(A3 α2 + B3 ) + , α2 (d2 α2 − d2 − α1 c2 ) α2 β2 (d2 α2 − d2 − α1 c2 ) r1 r2 r3 x∗ y ∗ z ∗ (d2 α2 − d2 − α1 c2 ), (c1 + x∗ + c2 y ∗ )2

M

a3 == where

ED

A1 = −r2 d2 , B1 = r2 d2 (1 + α1 c1 ) + r1 d1 (c1 d2 − c2 ) − r1 (d2 + β1 ),

PT

C1 = r1 [β1 (1 + α1 c1 ) + d2 (1 + c1 d1 ) + c2 (α1 − d1 ), ]

A2 = (d2 − β1 )(d2 + α1 c2 ) + (c1 d2 − c2 )(d1 d2 + α1 β1 ),

CE

B2 = (d2 + α1 c2 )2 + d1 (c1 d2 − c2 )(d2 + α1 c2 ) + β1 (1 + α1 c1 )(d2 + α1 c2 ), A3 = d2 + β1 , B3 = −d2 (1 + c1 d1 ) − c2 (α1 − d1 ) − β1 (1 + α1 c1 ).

AC

From Routh-Hurwitz criteria and conditions for Hopf bifurcation by Yu (2005),

we conclude the following result. Theorem 4.1. (i) If a1 > 0, a1 a2 − a3 > 0, a3 > 0, then the equilibrium P ∗ is locally asymptotically stable. 21

(4.3)

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(ii) If a1 < 0, a1 a2 − a3 < 0, or a3 < 0, then the equilibrium P ∗ is unstable. (iii) Hopf bifurcation occurs at P ∗ if and only if a1 a2 − a3 = 0.

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(4.4)

Theorem 4.1 demonstrates that P ∗ can only lose its stability through a Hopf bifurcation, which leads to periodic oscillations near P ∗ . Numerical simulations in Sections 5.1-5.3 of this work show that the Hopf bifurcation is possible. By (2.8) and (4.1), a direct computation shows

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x∗ − x¯+ = c2 y ∗ . Therefore, we conclude the following result.

Corollary 4.2. Assume a1 > 0, a1 a2 − a3 > 0. If conditions in Theorem 3.1 are satisfied, then system (2.3) is permanent and the unique interior equilibrium P ∗ is

M

asymptotically stable and satisfies x∗ − x¯+ > 0.

ED

Proof. Since conditions in Theorem 3.1 are satisfied, system (2.3) is permanent and then there is a unique interior equilibrium P ∗ with x∗ − x¯+ > 0. Thus we need to show that P ∗ is asymptotically stable, i.e., a3 > 0 by Theorem 4.1(i). (2)

(3)

(2)

PT

From λ1 > 0, λ1 > 0, λ13 > 0, we have

CE

α1 > d1 , α2 > 1 + c1 d1 , α2 < 1 + α1 c1 .

AC

From y ∗ > 0 and α2 < 1 + α1 c1 , we obtain α2 > α20 , which implies a3 > 0.

4.2

A set of equilibria

In this subsection, we consider a set of interior equilibria of system (2.3), in which there exist multiple stable equilibria. An interesting observation from simulations is that when the set of equilibria disappears, system (2.3) admits a stable periodic orbit.

22

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We assume c1 d2 − c2 = 0 in this section. Then we have α20 = 1 + α1 c1 . When

α2 = α20 and x¯+ < x+ , a direct computation shows that there is a set S of interior equilibria P (x, y, z) of (2.3), which can be described by α1 α2 x(x+ − x) (x − x¯+ ), z = }. d2 d2 β2 (d1 d2 + α1 β1 )

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S = {(x, y, z) : x ∈ (¯ x+ , x+ ), y =

We can see that if x = x¯+ , then y = 0, z = z + . If x = x+ , then y = y + , z = 0. Thus, the set S is an arc that connects equilibria P12 and P13 in the (x, y, z)-space.

From a3 = 0, equilibrium P ∈ S has a zero eigenvalue. Since P12 has two eigenvalues with negative real parts, it follows from continuity of eigenvalues with

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equilibria, P ∈ S that is near P12 also has two eigenvalues with negative real parts, which implies that P is stable. On the other hand, when α2 6= α20 , it follows from the expression of x∗ and c1 d2 − c2 = 0 that there is no interior equilibrium. Therefore, we conclude the following result.

Theorem 4.3. Assume c1 d2 − c2 = 0 and x¯+ < x+ . If α2 = α20 , there is a set S

M

of interior equilibria of (2.3) that connects equilibria P12 and P13 , and the equilibria

ED

near P12 is stable. Otherwise, there is no interior equilibrium. In the following simulations, we fix r1 = 1, r2 = r3 = 0.1, d1 = d2 = 0.01, c1 = 19, c2 = 0.19, α1 = 0.03, β1 = β2 = 0.01, and let α2 vary. Then we have c1 d2 = c2

PT

and α20 = 1.57. We will display an interesting bifurcation phenomenon in system (2.3): When α2 is small, there is no interior equilibrium of (2.3) and all positive

CE

solutions converge to the boundary equilibrium P12 ; When α2 is intermediate, there is a set of interior equilibria S and some of them are stable; When α2 becomes

AC

large, S disappears and all positive solutions converge to an interior periodic orbit ψ; However, when α2 is extremely large, ψ disappears and all positive solutions converge to a boundary periodic orbit φ. When α2 = 1.56 < α20 is small, simulations show that positive solutions of (2.3)

satisfy z → 0, as shown in Fig. 1a. Since α1 > d1 , P12 is globally asymptotically stable on the (x, y)-plane. Thus the solutions will converge to P12 . When α2 = 1.57 = α20 is intermediate, system (2.3) has a set of interior equilibria

23

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S, as shown in Fig. 1b: y = 100 − 3x, z = 6.28x(50 − x),

100 < x < 50. 3

Equilibria on S are divided into two parts: the equilibria near P12 are stable, while

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those near P13 are unstable, as shown in red and black curves in Fig. 1b. Indeed, when z → 0+ on S, we have x → x+ and y → y + . A direct computation shows that a1 > 0, a2 > 0, a3 = 0. From Theorem 4.1, the equilibria near P12 are stable. Similarly, when y → 0+ on S, we have x → x¯+ and z → z + , and a1 < 0, a2 > 0, a3 = 0. From Theorem 4.1, the equilibria near P13 have two eigenvalues with

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positive real parts and are unstable. Moreover, simulations show that there is no periodic oscillation and different solutions converge to different equilibria. When α2 = 1.595 > α20 is large, we have y ∗ < 0 and then system (2.3) has no interior equilibrium. Simulations show that there is a stable periodic orbit ψ, as shown in Fig. 1c and Fig. 2.

M

When α2 = 1.8  α20 is extremely large, simulations show that y → 0 and positive solutions converge to the periodic orbit φ on the (x, z)-plane, as shown in

ED

Fig. 1d.

Remark 4.4. Theoretical analysis in this section demonstrates that crow-cuckoo interaction can be mutualism when the three species coexist. First, as shown in

PT

Corollary 4.2, when system (2.3) is permanent and the interior equilibrium P ∗ is asymptotically stable, the crow-cuckoo interaction is mutualism at P ∗ . The under-

CE

lying reason is that the predation (e.g., α2 ) and the deterrence (e.g., α1 ) are well

AC

balanced. Second, as shown in Theorem 4.3, at the stable equilibrium P near P12 , . the crow-cuckoo interaction is mutualism since x∗ − x¯+ = x+ − x¯+ > 0. Therefore, the results confirm that the crow-cuckoo interaction can change from parasitism to mutualism observed by Canestrari et al. (2014).

5

Transition of interaction

In this section, we demonstrate transition of crow-cuckoo interaction with parameters by numerical simulations. We focus on parameters α2 , β2 and α1 , while similar 24

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simulations can be given for others. As mentioned in Section 2 of this work, α2 and β2 represent the intensity of predation by cats, while α1 denotes the efficiency of cuckoos, which affects the deterrence to predation by determining the number of

5.1

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cuckoo chicks.

Transition with α2

In this section, we exhibit the transition when the efficiency of cats (α2 ) varies, as shown in Fig. 3.

We fix r1 = 1, r2 = r3 = 0.1, d1 = d2 = 0.01, c1 = 20, c2 = 0.1, α1 = 0.03,

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β1 = β2 = 0.01, and let α2 vary.

Fig. 3a displays that when α2 (= 1.3) is small, the crow-cuckoo interaction is parasitism, as shown in Theorem 3.6(i). Since α2 > 1 + c1 d1 (= 1.2), Proposition 2.3(ii) shows that crows and cats coexist in the absence of cuckoos and the crows approach a density of x¯+ (= 66.7). However, when there exist cuckoos, numerical simulations

M

show that cats are driven into extinction while crows and cuckoos coexist, as shown in Fig. 3a. The crows approach a density of 49.9726(< 66.7), which is less than

ED

that in the crow-cat subsystem. Thus, the existence of cuckoos leads to decrease of population density of crows and the crow-cuckoo interaction is parasitism (+ −). Fig. 3b exhibits that when α2 (= 1.51) is intermediate, the crow-cuckoo interac-

PT

tion is mutualism, as shown in Corollary 4.2. Since α2 > 1 + 2c1 d1 /(1 − c1 d1 )(= 1.5), Proposition 2.3(iii) shows that crows and cats coexist in periodic oscillations in the

CE

absence of cuckoos. Numerical simulations show that the mean density of crows is 40.7650 (see remarks 5.1). However, when there exist cuckoos, the three species

AC

coexist at a steady state and crows approach a density of 47.5517(> 40.7650), which is larger than that in the crow-cat subsystem, as shown in Fig. 3b. The reason is that under the intermediate predation, the deterrence by cuckoos is effective and then the net effect of cuckoos on crows is positive. Since cuckoos depend upon crows for survival, the crow-cuckoo interaction is mutualism (+ +). Fig. 3c shows that when α2 (= 1.6) is large, the crow-cuckoo interaction is parasitism, where Hopf bifurcation occurs as shown in Theorem 4.1(iii). Numerical

25

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simulations show that in the absence of cuckoos, crows and cats coexist in periodic oscillations and the mean density of crows is 39.3509. However, when there exist cuckoos, the three species coexist in periodic oscillations and the mean density of crows is 36.1573(< 39.3509), which is less than that in the crow-cat subsystem. The

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reason is that when the predation is intensive, the deterrence by cuckoos becomes less effective and the net effect of cuckoos on crows is negative. Since cuckoos depend upon crows for survival, the crow-cuckoo interaction is parasitism (+ −).

Fig. 3d demonstrates that when α2 (= 6.6) is extremely large, the crow-cuckoo interaction is neutralism, as shown in Theorem 3.5(i). Numerical simulations in Fig.

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3d show that in this situation, cuckoos are driven into extinction by cats and play no role in the growth of crows. The reason is that when the predation is extremely intensive, crows will be at a very low density and cannot provide sufficient resources for the survival of cuckoos, which leads to their extinction. Thus, the crow-cuckoo interaction is neutralism (0 0).

M

On the other hand, recall that when α2 = 1.51 (resp. α2 = 1.6), the crowcuckoo interaction is mutualism (+ +) (resp. parasitism (+ −)). It follows from

ED

the continuity of solutions of (2.3) on parameters, there exists a value α ˜ 2 (1.51 < α ˜ 2 < 1.6) such that the when α2 = α ˜ 2 , the crow-cuckoo interaction is commensalism (+ 0). Therefore, the crow-cuckoo interaction can transition among parasitism,

PT

commensalism, mutualism and neutralism as α2 varies. Remark 5.1. When solutions of system (2.3) are in periodic oscillations, we apply

AC

CE

the following formula to compute the mean density of crows (x) 1 x¯ = T

Z

0

T

n

. 1X x(t)dt = xi n i=1

in the final period. Here, T is the period. xi is the value of x in numerical com-

putations of solutions by MatLab and n is the number of x in the period. In our simulations by MatLab, the time interval of solutions in each program is [0 20000] and thus the numerical values of the solutions are stable.

26

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5.2

Transition with β2

In this section, we demonstrate the transition when the saturation level of cats (β2 ) varies, as shown in Fig. 4. We fix r1 = 1, r2 = r3 = 0.1, d1 = d2 = 0.01, c1 = 20, c2 = 0.1, α1 = 0.03, the crow-cuckoo system by Proposition 2.2.

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β1 = 0.01, e = 151, α2 = eβ2 , and let β2 vary. Since α1 > d1 , cuckoos can persist in Fig. 4a displays that when β2 (= 0.008) is small, the crow-cuckoo interaction is parasitism. Since α2 = e ∗ β2 = 1.2080 > 1 + c1 d1 (= 1.2). Proposition 2.3(ii) shows that crows and cats can coexist in the absence of cuckoos, and the crows approach a

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density of x¯+ (= 96.1539). However, when there exist cuckoos, simulations show that cats are driven into extinction while crows and cuckoos coexist, as shown in Fig. 4a. The crows approach a density of x+ (= 49.9726 < 96.1539), which is less than that in the crow-cat subsystem. Thus, the crow-cuckoo interaction is parasitism (+ −). Fig. 4b exhibits that when β2 (= 0.01) is intermediate, the crow-cuckoo inter-

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action is mutualism. Numerical simulations show that in the absence of cuckoos, crows and cats coexist in periodic oscillations and the mean density of crows is

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40.7650. However, when there exist cuckoos, the three species coexist at a steady state and crows approach a density of 47.5517(> 40.7650), which is larger than that

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in the crow-cat subsystem, as shown in Fig. 4b. Thus, the crow-cuckoo interaction is mutualism (+ +).

Fig. 4c shows that when β2 (= 0.0105) is large, the crow-cuckoo interaction

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is parasitism. Numerical simulations show that in the absence of cuckoos, crows and cats coexist in periodic oscillations and the mean density of crows is 40.6257.

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However, when there exist cuckoos, the three species coexist in periodic oscillations and the mean density of crows is 35.6162(< 40.6257), which is less than that in the crow-cat subsystem. Thus, the crow-cuckoo interaction is parasitism (+ −). Fig. 4d demonstrates that when β2 (= 0.02) is extremely large, the crow-cuckoo interaction is neutralism. Numerical simulations in Fig. 4d show that in this situation, cuckoos are driven into extinction by cats. Thus, the crow-cuckoo interaction is neutralism (0 0).

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On the other hand, recall that when β2 = 0.01 (resp.

β2 = 0.0105), the

crow-cuckoo interaction is mutualism (+ +) (resp. parasitism (+ −)). It follows from the continuity of solutions of (2.3) on parameters, there exists a value β˜2 (0.01 < β˜2 < 0.0105) such that when β2 = β˜2 , the crow-cuckoo interaction is

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commensalism (+ 0). Therefore, the crow-cuckoo interaction can change among parasitism, commensalism, mutualism and neutralism as β2 varies.

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Transition with α1

In this section, we exhibit the transition when the efficiency of cuckoos (α1 ) varies,

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as shown in Fig. 5.

We fix r1 = 1, r2 = r3 = 0.1, d1 = d2 = 0.01, c1 = 20, c2 = 0.1, α2 = 1.51, β1 = β2 = 0.01, and let α1 vary. Since α2 > 1 + 2c1 d1 /(1 − c1 d1 )(= 1.5), Proposition 2.3(ii) shows that crows and cats can coexist in periodic oscillations in the absence of cuckoos, where the mean density of crows is 40.7650.

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Fig. 5a displays that when α1 (= 0.06) is large, the crow-cuckoo interaction is parasitism. Indeed, when there exist cuckoos, numerical simulations show that cats

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are driven into extinction while crows and cuckoos coexist, as shown in Fig. 5a. The crows approach a density of x+ (= 28.5714 < 40.7650), which is less than that in the crow-cat subsystem. The reason is that with a large efficiency, the cuckoos

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can approach a large population density, which forms a strong competition with crows. Since cuckoos depend upon crows for survival, the crow-cuckoo interaction

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is parasitism (+ −).

Fig. 5b exhibits that when α1 (= 0.03) is intermediate, the crow-cuckoo inter-

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action is mutualism. Indeed, when there exist cuckoos, the three species coexist at a steady state and crows approach a density of 47.5517(> 40.7650), which is larger than that in the crow-cat subsystem, as shown in Fig. 5b. The reason is that with an intermediate efficiency, the cuckoos can approach an intermediate population density, which can effectively deter the visitation of cats and will not form a strong competition with crows. Thus, the net effect of cuckoos on crows is positive. Since cuckoos depend upon crows for survival, the crow-cuckoo interaction is mutualism

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(+ +). Fig. 5c shows that when α1 (= 0.0256) is small, the crow-cuckoo interaction is parasitism. Indeed, when there exist cuckoos, the three species coexist in periodic oscillations and the mean density of crows is 39.4354(< 40.7650), which is less than

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that in the crow-cat subsystem. The reason is that with the small efficiency, the cuckoos approach a small population density, which cannot effectively deter the visitation of cats. Since cuckoos depend upon crows for survival, the crow-cuckoo interaction is parasitism (+ −).

Fig. 5d demonstrates that when α1 (= 0.011) is extremely small, the crow-

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cuckoo interaction is neutralism. Numerical simulations in Fig. 5d show that in this situation, cuckoos are driven into extinction by cats and play no role in the growth of crows. The reason is that with an extremely small efficiency, the cuckoos cannot survive in the presence of cats who kill crows and then lead to decrease of resources for cuckoos. Thus, the crow-cuckoo interaction is neutralism (0 0).

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On the other hand, recall that when α1 = 0.03 (resp. α1 = 0.0256), the crowcuckoo interaction is mutualism (+ +) (resp. parasitism (+ −)). It follows from the

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continuity of solutions of (2.3) on parameters, there exists a value α ˜ 1 (0.0256 < α ˜1 < 0.03) such that the when α1 = α ˜ 1 , the crow-cuckoo interaction is commensalism (+ 0). Therefore, the crow-cuckoo interaction can transition among parasitism,

Discussion

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commensalism, mutualism and neutralism as α1 varies.

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In this paper, we consider transition of interaction between a cuckoo and its host. Based on the analysis of a crow-cuckoo-predator(cat) model, we demonstrate mechanism by which the crow-cuckoo interaction changes with parameters. Results in this work not only confirm those exhibited in empirical observations

and experimental findings in a previous work, but also predict new phenomena. First, this work confirms the unexpected transition in which the crow-cuckoo interaction can change from parasitism to mutualism as the intensity of predation increases (Canestrari et al. 2014). Indeed, in sections 5.1-2 of this work, the transi29

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tion is confirmed as shown in Figs. 3-4, diagrams a-b. Moreover, a new phenomenon is revealed in Section 5.3 that this transition can also occur when the efficiency of cuckoos in converting the parasitism into fitness decreases, as shown in Fig. 5a-b. Second, this work predicts new phenomena not reported by Canestrari et al. (2014).

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Indeed, in sections 5.1-3, it is exhibited that when the predation is extremely strong, the crow-cuckoo interaction will return from mutualism to predation, or change from mutualism to neutralism, as shown in Figs. 3-5, diagrams c-d.

Varying parameters in system (2.3) can lead to transition of the crow-cuckoo interaction. We focus on the intensity of predation β2 , while similar discussions can

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be given for other parameters. As shown in sections 5.2 of this work, when the predation is weak, the deterrence by cuckoo chicks has a small positive effect on crows. Since the chicks consume resources in crow nests, the net effect of cuckoos on crows is negative and the crow-cuckoo interaction is parasitism. When the predation is intermediate, the deterrence by cuckoo chicks has a strong positive effect on crows

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and the net effect of cuckoos on crows becomes positive. Then the crow-cuckoo interaction is mutualism. However, when the predation is intensive, crows are at a

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low density, which leads to the decrease of cuckoos as shown in Figs. 4b-c. Thus, the effect of deterrence by cuckoo chicks becomes small and the net effect of cuckoos on crows becomes negative. Then the crow-cuckoo interaction returns to parasitism,

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as shown in Fig. 4c. When the predation is extremely intensive, crows are at a very low density and cannot provide sufficient resources for the survival of cuckoos, which

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leads to their extinction as shown in Fig. 4d. Thus, the crow-cuckoo interaction is neutralism.

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Population densities are important to the transition of crow-cuckoo interaction. As discussed in Theorem 3.6, a small initial density can lead to extinction of cuckoos y (or cats z). Moreover, as shown in Fig. 1b, solutions with different initial values converge to different equilibria, which correspond to different outcomes. Thus, varying initial population densities can lead to the transition of crow-cuckoo interaction among parasitism, commensalism, mutualism and neutralism. In the paper, two interesting phenomena are derived from dynamics of the model.

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One is that the crow-cuckoo interaction would return from mutualism to parasitism when the predation is extremely strong, while the other is that periodic oscillations of the population may occur in certain range of the parameters. Both of them may actually happen in nature. First, the change from mutualism to parasitism is

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possible. The reason is that when the predation is extremely strong, more crows are killed, which decreases the growth rate of cuckoo chicks. Thus the deterrence to predators is weakened and the crow-cuckoo interaction may change from mutualism to parasitism. Second, the periodic oscillations may actually occur. The reason is as follows. When crows are in high density, cuckoos and predators would grow

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rapidly since both cuckoos and predators are consumers of crows. The increase of the consumers would lead to the decrease of crows. Then the decrease of crows results in the decrease of both consumers because both of them depend on crows for survival. The decrease of the consumers is beneficial to the growth of crows, which leads to the increase of crows in return. This looks like an oscillating system.

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Numerical computations show that our model well fits the empirical observations by Canestrari et al.(2014). By linear regression method on MatLab, we ob-

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tain the parameter values of (2.3): r1 = 3.8881, r2 = 0.0001, r3 = 0.2678, r1 d1 = 1.1238, r2 d2 = 2.537, c1 = 0.0148, c2 = 2.8891, r1 β1 = r1 β2 = 0.1, r2 α1 = r3 α2 = 0.09 with initial value (2.9988, 1.6352, 18.6139). Numerical computations in Fig. 6 dis-

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plays that model (2.3) fits the empirical observations well. In addition, computations in Fig. 6 exhibit that x∗ = 3.4513 > xˆ = 3.3712, which implies that the crow

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and cuckoo are mutualistic. Thus, our result is consistent with the observation by Canestrari et al.(2014).

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In our simulations on the intensity of predation (β2 ) in Fig. 4, we use α2 = eβ2 . If we let β2 vary independently, a similar result can be shown. Although the crowcuckoo-predator model in this work is simple, it demonstrates mechanisms by which varying the environmental context can lead to transition of crow-cuckoo interaction from parasitism to mutualism, which would be helpful for understanding complexity in interspecific mutualisms.

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Acknowledgements. I am grateful to the three anonymous reviewers for their careful reading, helpful comments and suggestions that really helped me to improve the presentation of the paper. This work was supported by NSF of Guangdong

References

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(S2012010010320) and NSF of P.R. China (11571382).

[1] Bolopo D., Canestrari D., Roldan M., Baglione V. and Soler M. 2015. High begging intensity of great spotted cuckoo nestlings favours larger-size crow nest

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mates. Behav Ecol Sociobiol. 69:873-882.

[2] Bronstein J.I. 1994. Conditional outcomes in mutualistic interactions. Trends Ecol. Evol., 9:214-217.

[3] Butler G.J., Freedman H.I. and Waltman P. 1986. Uniformly persistent systems.

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Proc. Amer. Math. Sco. 96: 425-430.

[4] Butler G.J. and Waltman P. 1986. Persistence in dynamical systems. J. Differ-

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ential Equations 63: 255-263.

[5] Canestrari D., Bolopo D., Turlings T.C.J., Roder G., Marcos J.M. and Baglione

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V. 2014. From parasitism to mutualism: unexpected interactions between a cuckoo and its host. Science 343: 1350-1352.

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[6] Cantrell R.S., Cosner C. and Ruan S. 2004. Intraspecific interference and consumer-resource dynamics. Discrete and and Continuous Dynamical Systems-

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B. 4: 527-546.

[7] Cosner C. 1996. Variability, vagueness and comparison methods for ecological models. Bull. Math. Biol. 58: 207-246.

[8] Freedman H.I. and Moson P. 1990. Persistence definitions and their connections, Proc. Amer. Math. Soc. 109:1025-1033.

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[9] Freedman H.I., Ruan S. and Tang M. 1994. Uniform persistence and flows near a closed positively invariant set, J. Dynam. Differential Equations 6: 583-600. [10] Freeze M., Chang Y., Feng W. 2014. Analysis of dynamics in a complex food chain with ratio-dependent functional response. J. Appl. Anal. Comput.

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4(1):69-87.

[11] Hofbauer J. and Sigmund K., 1998. Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK.

[12] J.N. Holland, D.L. DeAngelis, 2010. A consumer-resource approach to the

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density-dependent population dynamics of mutualism. Ecology, 91: 1286-1295. [13] Hsu S.-B., Hwang T.-W. and Kuang Y. 2001. Rich dynamics of a ratiodependent one-prey two-predator model. J. Math. Biol. 43: 377-396. [14] Hsu S.B., Ruan S.,Yang T.H. 2013. On the dynamics of two-consumers-one-

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resource competing systemswith Beddington-DeAngelis functional response. Discrete Cont Dyn-B 18(9):2331-2353.

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[15] Hutson V. 1984. A theorem on average Liapunov functions. Monatshefte fur

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Mathematik 98:267-275.

[16] Kuang Y. and Freedman H. I. 1988. Uniqueness of limit cycles in Gause-type

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predator-prey systems. Math. Biosci. 88: 67-84. [17] Kang Y. and Wedekin L. 2013. Dynamics of a intraguild predation model with

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generalist or specialist predator. J. Math. Biol. 67(5):1227-59. [18] Neuhauser C. and Fargione J.E. 2004. A mutualism-parasitism continuum model and its application to plant-mycorrhizae interactions. Ecological Modelling 177: 337C352. [19] Ruer G., Canestrari D., Bolopo D. Marcos J., Villard N., Baglione V. and Turlings T. 2014. Chicks of the great spotted cuckoo may turn brood parasitism

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into mutualism by producing a foul-smelling secretion that repels predators. J. Chem. Ecol. 40:320-324. [20] Shu H., Hu X., Wang L. and Watmough J. 2015. Delay induced stability switch, 71: 1269-1298.

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multitype bistability and chaos in an intraguild predation model. J. Math. Biol.

[21] Verdy A., Amarasekare P. 2010. Alternative stable states in communities with intraguild predation. J Theor Biol 262(1):116-128.

[22] Wang Y. and DeAngelis, D.L. 2011. Transitions of interaction outcomes in a

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uni-directional consumer-resource system. J. Theoretical Biology. 280: 43-49.

[23] Wang Y. and DeAngelis, D.L. 2016. Stability of an intraguild predation system with mutual predation. Commun. Non. Sci. Nume. Simu. 33: 141-159. [24] Wang Y., DeAngelis D.L. and Holland J.N. 2015. Dynamics of an ant-plant-

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pollinator model. Commun. Non. Sci. Nume. Simu. 20: 950-964. [25] Yu P. 2005. Closed-form conditions of bifurcation points for general differential

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equations. Int J Bifurcation Chaos. Appl. Sci. Eng. 15:1467-1483.

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(a) Extinction of cats (z) as α =1.56

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Figure 1: Bifurcations of system (2.3) at a set of equilibria. For simplicity, we display variables x and z and omit y. Fix r1 = 1, r2 = r3 = 0.1, d1 = d2 = 0.01, c1 = 19, c2 = 0.19, α1 = 0.03, β1 = β2 = 0.01, and let α2 vary. We use the

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replacement of z := z/100 since z is large. In Fig. 5a, α2 (= 1.56) is small. All positive solutions of (2.3) converge to equilibrium P12 on the (x, y)-plane with z → 0.

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In Fig. 5b, α2 (= 1.57) is intermediate. There exist a set of interior equilibria S

which can be divided into two parts: the red one is stable, while the black one is unstable. All positive solutions (except the unstable equilibria) converge to the stable equilibria. In Fig. 5c, α2 (= 1.595) is large. system (2.3) has no interior equilibrium but admits a positive periodic solution, which is asymptotically stable. The three-dimensional figure is shown in Fig. 6. In Fig. 5d, α2 (= 1.8) is extremely large and all positive solutions of (2.3) converge to the periodic solution on the (x, z)-plane.

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each diagram represent population densities of crows, cuckoos and cats respectively, while we use the replacement of z := z/100 since z is large. Fix r1 = 1, r2 = r3 = 0.1,

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d1 = d2 = 0.01, c1 = 20, c2 = 0.1, α1 = 0.03, β1 = β2 = 0.01, and let α2 vary. In Fig. 3a, the efficiency is low (α2 = 1.3). Cats are driven into extinction by cuckoos and crows approach a density less than that in the crow-cat system. Thus the

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crow-cuckoo interaction is parasitism. In Fig. 3b, the efficiency is intermediate (α2 = 1.51). The three species coexist while crows approach a density larger than that in the crow-cat system. Thus the crow-cuckoo interaction is mutualism. In Fig. 3c, the efficiency is large (α2 = 1.6). The three species coexist while crows approach a density less than that in the crow-cat system. Thus the crow-cuckoo interaction is parasitism. In Fig. 3d, the efficiency is extremely large (α2 = 6.6). Cuckoos are driven into extinction by cats, which means that the crow-cuckoo interaction is neutralism. 37

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(a) Parasitism at β2=0.008

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d1 = d2 = 0.01, c1 = 20, c2 = 0.1, α1 = 0.03, β1 = 0.01, e = 151, and let β2 vary. In Fig. 4a, the predation is weak (β2 = 0.008). Cats are driven into extinction by cuckoos and crows approach a density less than that in the crow-cat system. Thus

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the crow-cuckoo interaction is parasitism. In Fig. 4b, the predation is intermediate (β2 = 0.01). The three species coexist while crows approach a density larger than that in the crow-cat system. Thus the crow-cuckoo interaction is mutualism. In Fig. 4c, the predation is large (β2 = 0.0105). The three species coexist while

crows approach a density less than that in the crow-cat system. Thus the crowcuckoo interaction is parasitism. In Fig. 4d, the predation is extremely large (β2 = 0.02). Cuckoos are driven into extinction by cats, which means that the crow-cuckoo interaction is neutralism. 38

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(a) Parasitism at α1 =0.06

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Figure 5: Solutions of system (2.3) when α1 varies. The red, blue and black lines in each diagram represent population densities of crows, cuckoos and cats respectively, while we use the replacement of z := z/100 since z is large. Fix r1 = 1, r2 = r3 = 0.1,

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d1 = d2 = 0.01, c1 = 20, c2 = 0.1, α2 = 1.51, β1 = β2 = 0.01, and let α1 vary. In Fig. 5a, α1 (= 0.06) is large. Cats are driven into extinction by cuckoos and crows

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approach a density less than that in the crow-cat system. Thus the crow-cuckoo interaction is parasitism. In Fig. 5b, α1 (= 0.03) is intermediate. The three species coexist while crows approach a density larger than that in the crow-cat system. Thus the crow-cuckoo interaction is mutualism. In Fig. 5c, α1 (= 0.0256) is small. The three species coexist while crows approach a density less than that in the crow-cat system. Thus the crow-cuckoo interaction is parasitism. In Fig. 5d, α1 (= 0.011) is extremely small. Cuckoos are driven into extinction by cats, which means that the crow-cuckoo interaction is neutralism. 39

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solutions of x, y and z are denoted by red, blue and black curves, respectively. Let r1 = 3.8881, r2 = 0.0001, r3 = 0.2678, r1 d1 = 1.1238, r2 d2 = 2.537, c1 = 0.0148, c2 = 2.8891, r1 β1 = r1 β2 = 0.1, r2 α1 = r3 α2 = 0.09 with initial value

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(2.9988, 1.6352, 18.6139). Numerical computations display that model (2.3) fits the real data well, in which the crow and cuckoo are mutualistic when there exist cats.

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