- Email: [email protected]
The effect of prey refuge in a patchy predator–prey system
Mathematical Biosciences 243 (2013) 126–130
Contents lists available at SciVerse ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs
The effect of prey refuge in a patchy predator–prey system q Zhihui Ma a, Shufan Wang b, Weide Li a, Zizhen Li a,⇑ a b
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030, China
a r t i c l e
i n f o
Article history: Received 9 May 2012 Received in revised form 6 November 2012 Accepted 21 February 2013 Available online 5 March 2013 Keywords: Patchy predator–prey model Migration Prey refuge Global stability Limit cycle
a b s t r a c t In this work, we proposed a patchy predator–prey model with one patch as refuge and the other as open habitat, and incorporated prey refuge in the considered model explicitly. We applied an analytical approach to study the dynamic consequences of the simplest forms of refuge used by prey and the migration efficiency. The results have shown that the refuge used by prey and the migration efficiency play an important role in the dynamic consequences of the interacting populations and the equilibrium density of two interacting populations. This work also proposed a new approach which can incorporate prey refuge in predator–prey system explicitly. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The impact of prey refuge on predator–prey systems is a fundamental and popular issue in applied mathematics and theoretical ecology. Most of theoretical work lead to the conclusion that refuge used by prey has a stabilizing effect on predator–prey systems [1– 15]. In fact, the consequences of refuge type for stability properties depend on the underlying model, but the general result is that refuge which protect a constant number of prey lead to a stable equilibrium and have a stronger stabilizing effect on the dynamic consequences of two interacting populations than refuge which protect a constant proportion of prey population [12,13]. The traditional ways in which the effect of refuge used by prey is to modify the functional response of predators and study the following general predator–prey model with prey refuge implicitly
x _ puðx xR Þy xðtÞ ¼ rx 1 K _ yðtÞ ¼ ðquðx xR Þ dÞy
ð1Þ
in which xðtÞ and yðtÞ denote the density of prey and predator populations at any time t respectively, and r; K; p; q, d are all positive constants and have its biological meanings accordingly. r is the intrinsic per capita growth rate of prey population, K is the environmental carrying capacity of prey population, p is the maximal per capita consumption rate of predators, d is the per capita death rate q This work was supported by the National Natural Science Foundation of China (No. 11126183). ⇑ Corresponding author. Tel.: +86 09318912483. E-mail address: [email protected] (Z. Li).
0025-5564/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.mbs.2013.02.011
of predators, and q is the efficiency with which predators convert consumed prey into new predators. xR represents a quantity of prey population that occupied a refuge. The term uðxÞ represents the functional response of the predator population and satisfies the following assumption:
uð0Þ ¼ 0; u0 ðxÞ > 0 ðx > 0Þ: Ma et al. [18] investigated the effect of prey refuge with regard to the local stability of the interior equilibrium point and the values of the equilibrium density of the above model. they obtained that the effect of refuge used by prey can stabilize the interior equilibrium point and destabilize it under a very restricted set of conditions which is disagreement with previous results in this field. Motivated by this, we propose a predator–prey model incorporating the effect of prey refuge explicitly in a two-patch environment (one patch represents an open habitat while the other is a refuge for prey) and will study theoretically its stabilizing impact on the dynamic consequences of the proposed model.
2. Mathematical model We assume that there are two habitats (patch 1 and patch 2) for predator and prey populations feeding on them. The patch 1 is an open habitat while the patch 2 is a completely spatial refuge on which prey population can survive only. Therefore, patch 2 can consist of separate patches for prey and predator populations. In order to form a predator-pry model incorporating prey refuge explicitly, we have the following assumptions
127
Z. Ma et al. / Mathematical Biosciences 243 (2013) 126–130
(1) Assuming that the population growth of prey is Logistic in patch 1 in absence of predation. The net growth (without migration) in patch 2 is negative due to lack of resources and mating opportunities. (2) Supposing that migrations between the patch 1 and the patch 2 are much more rapid than the other processes in the system like growth, maturity and predation. The parameter b is the migration rate from the former to the latter while a is the inverse rate (a þ b 6 1). According to the above assumptions, we can formulate the following predator–prey system incorporating the effect of refuge used by prey explicitly
dx1 x1 cx1 y ¼ ðax2 bx1 Þ þ e r 1 x1 1 d1 x1 a þ x1 ds K1 dx2 ¼ ðbx1 ax2 Þ þ eðd2 x2 Þ ds dy ecx1 ¼e d y ds a þ x1
ð2Þ
where x1 ðtÞ and x2 ðtÞ represent the density of prey population in patch 1 and 2, respectively, yðtÞ denotes the density of predators in patch 1. The parameters r1 ; K 1 ; c; di ði ¼ 1; 2Þ; d; e and a are all positive constants and have its biological meanings accordingly. r 1 is the intrinsic per capita growth rate of prey population in patch 1. K 1 is the common environmental carrying capacity. c is the attack rate of predators to prey population for patch 1. di is the mortality rate of prey population in patch i. d is the mortality rate of predators in patch 1. e is the efficiency with which predators convert conx1 sumed prey into new predators. The term aþx denotes the func1 tional response of the predator population in which a is the half saturation constant. This response function is termed as Holling II functional response. We choose the total densities to represent the system globally, that is, the aggregated variables are
x ¼ x1 þ x2 ;
y ¼ y;
_ xðtÞ ¼ _ yðtÞ ¼
bx
aþb
;
x ; 1 þ ab
aða þ bÞ þ ax
ð4Þ
dy
3. Positivity and boundedness
Theorem 3.1. All the solutions of model (4) which start in R2þ are positive and uniformly bounded. Proof. The positively of the solutions of model (4) can be easily obtained according to their ecological meanings. Next, we will only show the boundedness. Defining the function WðtÞ ¼ xðtÞ þ 1e yðtÞ. Hence, we have
a2 r 1 x d _ ¼ ar 1 bd2 x 1 W y: K 1 ðar 1 bd2 Þða þ bÞ e aþb Now, for each V > 0, we get
_ þ VW 6 K 1 ðar 1 bd2 Þða þ bÞ W 4a2 r 1 ðar 1 bd2 Þ
2
ar1 bd2 2 1 ðd VÞ: Vþ e aþb
Let us choose V > d, then the right hand is positive. Thus, the right hand side is bounded for all ðx; yÞ 2 R2þ . _ þ VW < U. Hence, we choose a U such that W Applying the theory of differential inequality, we obtain that
0 < Wðx; yÞ <
U U 1 eVt þ W ðxð0Þ; yð0ÞÞeVt ! ; V V
t ! þ1:
Therefore, all the solutions of model (4) starting in R2þ are confined to the region D, where
D¼
ðx; yÞ 2 R2þ jW ¼
U þ e; e > 0 : V
h
4. The equilibria
x1 ¼ x x2 ;
The possible equilibria of model (4) can be obtained by solving the following equations
or
x1 ¼
aecxy
In the following sections, we will consider the dynamic behaviors of the above aggregated model.
x is invariant for the fast part. Migration changes the proportions of prey on the two patches, but not their total density x. The solutions of the fast part is described as follows
x2 ¼
ðar1 bd2 Þx a2 r1 x2 acxy aþb K 1 ða þ bÞ2 aða þ bÞ þ ax
x2 ¼ x x1 :
b The parameter aþb represents a constant proportion of prey population staying on patch 2 and denotes the constant proportion of prey using refuge. The parameter ab is the ratio of the migration rate from patch 1 to patch 2 to that from patch 2 to patch 1 and represents the migration efficiency. Therefore, the aggregated model can be obtained by following steps.
aecxy aða þ bÞ þ ax
We can easily calculate all equilibria of model (4) are P0 ð0; 0Þ,
adða þ bÞ ; aðec dÞ
and P2 ðx ; y Þ, where
y ¼
er1 a2 r1 x x 1 : d K 1 ðar1 bd2 Þða þ bÞ
The equilibrium point P2 ðx ; y Þ is positive if and only if
b ð3Þ
(II) Substituting x1 and x2 into model (3), we obtain the aggregated system at the slow time scale
ð5Þ
dy ¼ 0
2 ÞðaþbÞ P1 ðK 1 ðar1 bd ; 0Þ a2 r1
x ¼
(I) Adding the first two equations of model (2), we have
dx1 x1 cx1 y d1 x1 d2 x2 ¼ e r 1 x1 1 a þ x1 dt K1 dy ecx1 ¼e d y ds a þ x1
ðar1 bd2 Þx a2 r1 x2 acxy ¼0 aþb K 1 ða þ bÞ2 aða þ bÞ þ ax
aþb
<1
K 1 d2 ðec dÞ ; K 1 ðr 1 þ d2 Þðec dÞ adr1
or
b
a
<
r1 ad : 1 K 1 ðec dÞ d2
128
Z. Ma et al. / Mathematical Biosciences 243 (2013) 126–130
Obviously, the equilibrium density of prey population x increases as the proportion of prey refuge increases according to b the representation of x with respect to aþb and increases with b the ratio a. On the other hand, one can easily see that y is a continb uous function of parameter aþb , simple computation shows that
dy aer1 ¼ b d aþb ðec dÞ 1
8 2 32 9 b < = 1 aþb adr1 ðr 1 þ d2 Þ 4 5 : 2 :1 b b K 1 ðec dÞ 1 aþb d2 ; r 1 aþb
b aþb
Hence, considering into two cases qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 ðecdÞ 1 adr 1 ðr1 þd2 Þ
b Case 4.1. If 0 < aþb < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K 1 ðecdÞ ðr þd2 Þ1 adr 1 ðr 1 þd2 Þ 1
, then y is a strictly increas-
ing function. That is, increasing the amount of prey refuge can increase the equilibrium density of predators. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 ðecdÞ 1 adr 1 ðr 1 þd2 Þ
Case 4.2. If qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K 1 ðecdÞ ðr þd2 Þ1 adr 1 ðr1 þd2 Þ 1
b < aþb , then y is a strictly decreasing
function. That is, increasing the proportion of prey refuge will decrease the equilibrium density of predators.
Again, we can easily see that y is a continuous function of parameter ab and simple computation obtains that
" # dy aer1 adr1 ðr1 þ d2 Þ ¼ 1 2 : ec d d ab K 1 ðec dÞ r 1 b d2 a
Similarly, we consider into two cases qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þd2 Þ , then y is a strictly increasCase 4.3. If 0 < ab < dr12 ½1 Kadðr 1 r 1 ðecdÞ ing function. That is, increasing the amount of prey refuge can increase the equilibrium density of predators. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þd2 Þ < ab, then y is a strictly decreasing Case 4.4. If dr12 ½1 Kadðr 1 r 1 ðecdÞ function. That is, increasing the proportion of prey refuge will decrease the equilibrium density of predators.
5. Stability analysis
J1 ¼ @
or
b
a
>
r1 ad : 1 K 1 ðec dÞ d2
Otherwise, it is locally unstable and the equilibrium point P2 ðx ; y Þ lies in fðx; yÞjx > 0; y < 0g. Again, the Jacobian matrix for the interior equilibrium point P2 ðx ; y Þ is as follows
J2 ¼
a11
a12
a21
0
where
r1 a2 r1 ðec þ dÞx d ; ec K 1 ðar 1 bd2 Þða þ bÞ aða þ bÞaecy ¼ > 0: ½aða þ bÞ þ ax 2
a11 ¼ a21
a12 ¼
cx < 0; a þ x
Clearly, DetJ 2 ¼ a12 a21 > 0, hence, the sign of the eigenvalues depends only on trJ 2 ¼ a11 . Therefore, the interior equilibrium point P 2 ðx ; y Þ is locally asymptotically stable if and only if a11 < 0, that is
b
aþb
>1
K 1 d2 ðec dÞ ; K 1 ðr 1 þ d2 Þðec dÞ ar1 ðec þ dÞ
or
b
a
>
r1 aðec þ dÞ 1 : K 1 ðec dÞ d2
Otherwise, the interior equilibrium point is locally unstable. Next, we will prove the global stability of the interior equilibrium point. Let us consider model (4) in the following form
_ xðtÞ ¼ xgðxÞ ypðxÞ _ yðtÞ ¼ yðqðxÞ dÞ
ð6Þ
Where
In this section, we will consider the stability properties of the equilibria of model (4). Clearly, the equilibrium point P 0 ð0; 0Þ is a saddle point and is unstable. The Jacobian matrix of model (4) at the predator-free equilib2 ÞðaþbÞ rium point P 1 ðK 1 ðar1 bd ; 0Þ is given by a2 r
0
That is
b K 1 d2 ðec dÞ ; >1 K 1 ðr 1 þ d2 Þðec dÞ adr1 aþb
1
1
r
1 ðar 1 bd2 Þ aarcK1 þK 1 ðar1 bd2 Þ
0
ecK 1 ðar 1 bd2 Þ aar1 þK 1 ðar1 bd2 Þ d
A
2 ÞðaþbÞ Thus, The predator-free equilibrium point P1 ðK 1 ðar1 bd ; 0Þ is a2 r1 locally asymptotically stable if and only if
ecK 1 ðar 1 bd2 Þ d < 0: aar1 þ K 1 ðar1 bd2 Þ
0
p ðxÞ 0 d xg ðxÞ þ gðxÞ xgðxÞ pðxÞ dx qðxÞ d
!
gðxÞ ¼
ar1 bd2 a2 r1 x acx ; pðxÞ ¼ ; qðxÞ ¼ epðxÞ: aða þ bÞ þ ax aþb K 1 ða þ bÞ2
We have the following Lemma regard uniqueness of limit cycles of model (4). Lemma 5.1. Supposing in model (6)
! p0 ðxÞ 0 d xg ðxÞ þ gðxÞ xgðxÞ pðxÞ 60 dx qðxÞ d in 0 6 x < x and x < x 6 K. Then model (6) has exactly one limit cycle which is globally asymptotically stable with respect to the region D ¼ fðx; yÞjx > 0; y > 0g n fE2 ðx ; y Þg.
3 2 ðar 1 bd2 Þa2 r 1 x ar1 bd2 a2 r1 x ar1 bd2 a2 r 1 x aa d 4 K 1 ðar1 bd2 ÞðaþbÞ2 þ aþb 1 K 1 ðar1 bd2 ÞðaþbÞ aþb 1 K 1 ðar1 bd2 ÞðaþbÞ aðaþbÞþax 5 6 0 () aecx dx d aðaþbÞþax " # aðaþbÞ K 1 ðar 1 bd2 ÞðaþbÞ d xð2x þ a a2 r1 6 0 () P 0: adðaþbÞ dx x aðecdÞ
Z. Ma et al. / Mathematical Biosciences 243 (2013) 126–130
According to Lemma 5.1, we obtain that It is equivalent to showing that
2 adða þ bÞ adða þ bÞ K 1 ðar 1 bd2 Þða þ bÞ aða þ bÞ x þ 2a2 r 2a aðec dÞ aðec dÞ 2 adða þ bÞ P 0: aðec dÞ aþbÞ 2 Since ðx adð aðecdÞ Þ > 0 and
adðaþbÞ aðecdÞ
> 0, the above inequality is
equivalent to the following form
adða þ bÞ K 1 ðar 1 bd2 Þða þ bÞ aða þ bÞ : < 2a2 r1 2a aðec dÞ That is
b
aþb or
b
a
<
<1
K 1 d2 ðec dÞ ; K 1 ðr1 þ d2 Þðec dÞ ar 1 ðec þ dÞ
r1 aðec þ dÞ : 1 K 1 ðec dÞ d2
Hence, model (4) has exactly one limit cycle which is globally stable if and only if
b
aþb or
b
a
<
<1
K 1 d2 ðec dÞ ; K 1 ðr1 þ d2 Þðec dÞ ar 1 ðec þ dÞ
r1 aðec þ dÞ 1 : K 1 ðec dÞ d2
Noticing that the nonexistence of limit cycles also implies the globally asymptotical stability of the interior equilibrium point of model (4). Therefore, we have the following main results 1 þaÞ Theorem 5.2. Suppose that e > dðK cðK 1 aÞ , we have
b 1 d2 ðecdÞ (1) If 0 < aþb < 1 K 1 ðr1 þdK2 ÞðecdÞar , then the predator and prey 1 ðecþdÞ
populations periodically oscillate around the unique equilibrium point in the first quadrant. K 1 d2 ðecdÞ b 1 d2 ðecdÞ < aþb < 1 K 1 ðr1 þd , (2) If 1 K 1 ðr1 þd2KÞðecdÞar 1 ðecþdÞ 2 ÞðecdÞadr 1
then
the two interacting populations tend to reach a globally asymptotically stable equilibrium point in the first quadrant. K 1 d2 ðecdÞ b < aþb < 1, then predators go extinct (3) If 1 K 1 ðr1 þd 2 ÞðecdÞadr 1
129
and incorporating the effect of refuge used by prey in the considered models implicitly. However, we present another approach allowing to propose a patchy predator–prey model with one patch as refuge and incorporate prey refuge in the predation model explicitly. The results show that the refuge used by prey plays an important role in interacting populations dynamic consequences and the equilibrium density of two interacting populations. Furthermore, the impact of the migration efficiency on the proposed model is also investigated. The analysis on the stability of the predator–prey system reveals the following conclusions 1 The equilibrium density of prey population increases as refuge used by prey and/or the migration efficiency increasing while that of predators decreases with prey refuge and/or the migration efficiency. Although several works have shown an increase in prey density as the refuge used by prey, the effect on predator density is less clear. Our results are consistent with those of lez-Olivars and Ramos-Jiliberto [13] and contrast with Gonza those of McNair [17] who predicted an increase in the equilibrium density of predator population with prey refuge. 2 When the effect of prey refuge and/or the migration efficiency are strong enough, the predators will extinct and the prey population reach their environmental carrying capacity eventually. This behavior also observed by Collings [16] and Ma et al. [18]. 3 Our results show that the effect of prey refuge and/or the migration efficiency play an important role in determining the stability of the interior equilibrium point of the considered model. The refuge used by prey and/or the migration efficiency can increase the global stability of the interior equilibrium point (stabilizing effect) which is agreement with most previous results on simple models [6,7,9,13–15] and disagreement with that of Ma et al. [18] who found a clear destabilizing effect on the considered model. In this paper, increase in global stability or stabilization refers to cases where an interior equilibrium point changes from repeller to an attractor due to changes in the values of the controlled parameter [See Ma et al. [18]]. Increase in stability of the interior equilibrium point of the considered model with increasing prey refuge is usually associated with predator population which exhibit saturating functional responses to prey population.
Acknowledgements
while prey population reaches its maximum environmental carrying capacity.
1 þaÞ , we obtain that Theorem 5.3. Suppose that e > dðK cðK 1 aÞ
(1) If 0 < ab < dr12 ½1 KaðecþdÞ , then the prey population and preda1 ðecdÞ tors periodically oscillate around the unique equilibrium point in the first quadrant. ad (2) If dr12 ½1 KaðecþdÞ < ab < dr12 ½1 K 1 ðecdÞ , then the two interacting 1 ðecdÞ populations tend to reach a globally asymptotically stable equilibrium point in the first quadrant. ad < ab, then the prey population reaches its max(3) If dr12 ½1 K 1 ðecdÞ imum environmental carrying capacity while predators go extinct. 6. Conclusion When considering the prey refuge in predator–prey systems, one usually concentrates on modifying the functional response
This work was supported by the National Natural Science Foundation of China (No. 11126183) and the Fundamental Research Funds for the Central Universities (No.lzujbky-2011-48). References [1] C.S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol. 91 (1959) 385. [2] M.P. Hassel, R.M. May, Stability in insect host–parasite models, J. Anim. Ecol. 42 (1973) 693. [3] J.M. Smith, Models in Ecology, Cambridge Univ Press, Cambridge, UK, 1974. [4] W.W. Murdoch, A. Stewart-Oaten, Predation and population stability, Adv. Ecol. Res. 9 (1975) 1. [5] M.P. Hassell, The Dynamics of Arthropod Predator–Prey Systems, Princeton Univ. Press, Princeton, NJ, 1978. [6] M.A. Hoy, Almonds (California), in: W. Helle, M.W. Sabelis (Eds.), Spider Mites: Their Biology, Natural Enemies and Control, World Crop Pests, vol. 1B, Elsevier, Amsterdam, 1985. [7] A. Sih, Prey refuges and predator–prey stability, Theor. Popul. Biol. 31 (1987) 1. [8] A.R. Ives, A.P. Dobson, Antipredator behavior and the population dynamics of simple predator–prey systems, Am. Nat. 130 (1987) 431. [9] G.D. Ruxton, Short term refuge use and stability of predator–prey models, Theor. Popul. Biol. 47 (1995) 1. [10] M.E. Hochberg, R.D. Holt, Refuge evolution and the population dynamics of coupled host–parasitoid associations, Evol. Ecol. 9 (1995) 633. [11] J. Michalski, J.C. Poggiale, R. Arditi, P.M. Auger, Macroscopic dynamic effects of migration in patchy predator–prey systems, J. Theor. Biol. 185 (1997) 459.
130
Z. Ma et al. / Mathematical Biosciences 243 (2013) 126–130
[12] V. Kurivan, Effects of optimal antipredator behvior of prey on predator–prey dynamics: the role of reufuges, Theor. Popul. Biol. 53 (1998) 131. [13] E. González-Olivars, R. Ramos-Jiliberto, Dynamics consequences of prey refuges in a simple model system: more prey, few predators and enhanced stability, Ecol. Model. 166 (2003) 135. [14] T.K. Kar, Stability analysis of a prey–predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul. 10 (2005) 681. [15] Y. Huang, F. Chen, L. Zhong, Stability analysis of a prey–predator model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput. 182 (2006) 672.
[16] J.B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator–prey interaction model incorporating a prey refuge, Bull. Math. Biol. 57 (1995) 63. [17] J.M. McNair, The effects of refuges on predator–prey interactions: a reconsideration, Theor. Popul. Biol. 29 (1986) 38. [18] Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang, Z. Li, Effects of prey refuges on a predator–prey model with a class of functional responses: the role of refuges, Math. Biosci. 218 (2009) 73.
Contents lists available at SciVerse ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs
The effect of prey refuge in a patchy predator–prey system q Zhihui Ma a, Shufan Wang b, Weide Li a, Zizhen Li a,⇑ a b
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030, China
a r t i c l e
i n f o
Article history: Received 9 May 2012 Received in revised form 6 November 2012 Accepted 21 February 2013 Available online 5 March 2013 Keywords: Patchy predator–prey model Migration Prey refuge Global stability Limit cycle
a b s t r a c t In this work, we proposed a patchy predator–prey model with one patch as refuge and the other as open habitat, and incorporated prey refuge in the considered model explicitly. We applied an analytical approach to study the dynamic consequences of the simplest forms of refuge used by prey and the migration efficiency. The results have shown that the refuge used by prey and the migration efficiency play an important role in the dynamic consequences of the interacting populations and the equilibrium density of two interacting populations. This work also proposed a new approach which can incorporate prey refuge in predator–prey system explicitly. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The impact of prey refuge on predator–prey systems is a fundamental and popular issue in applied mathematics and theoretical ecology. Most of theoretical work lead to the conclusion that refuge used by prey has a stabilizing effect on predator–prey systems [1– 15]. In fact, the consequences of refuge type for stability properties depend on the underlying model, but the general result is that refuge which protect a constant number of prey lead to a stable equilibrium and have a stronger stabilizing effect on the dynamic consequences of two interacting populations than refuge which protect a constant proportion of prey population [12,13]. The traditional ways in which the effect of refuge used by prey is to modify the functional response of predators and study the following general predator–prey model with prey refuge implicitly
x _ puðx xR Þy xðtÞ ¼ rx 1 K _ yðtÞ ¼ ðquðx xR Þ dÞy
ð1Þ
in which xðtÞ and yðtÞ denote the density of prey and predator populations at any time t respectively, and r; K; p; q, d are all positive constants and have its biological meanings accordingly. r is the intrinsic per capita growth rate of prey population, K is the environmental carrying capacity of prey population, p is the maximal per capita consumption rate of predators, d is the per capita death rate q This work was supported by the National Natural Science Foundation of China (No. 11126183). ⇑ Corresponding author. Tel.: +86 09318912483. E-mail address: [email protected] (Z. Li).
0025-5564/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.mbs.2013.02.011
of predators, and q is the efficiency with which predators convert consumed prey into new predators. xR represents a quantity of prey population that occupied a refuge. The term uðxÞ represents the functional response of the predator population and satisfies the following assumption:
uð0Þ ¼ 0; u0 ðxÞ > 0 ðx > 0Þ: Ma et al. [18] investigated the effect of prey refuge with regard to the local stability of the interior equilibrium point and the values of the equilibrium density of the above model. they obtained that the effect of refuge used by prey can stabilize the interior equilibrium point and destabilize it under a very restricted set of conditions which is disagreement with previous results in this field. Motivated by this, we propose a predator–prey model incorporating the effect of prey refuge explicitly in a two-patch environment (one patch represents an open habitat while the other is a refuge for prey) and will study theoretically its stabilizing impact on the dynamic consequences of the proposed model.
2. Mathematical model We assume that there are two habitats (patch 1 and patch 2) for predator and prey populations feeding on them. The patch 1 is an open habitat while the patch 2 is a completely spatial refuge on which prey population can survive only. Therefore, patch 2 can consist of separate patches for prey and predator populations. In order to form a predator-pry model incorporating prey refuge explicitly, we have the following assumptions
127
Z. Ma et al. / Mathematical Biosciences 243 (2013) 126–130
(1) Assuming that the population growth of prey is Logistic in patch 1 in absence of predation. The net growth (without migration) in patch 2 is negative due to lack of resources and mating opportunities. (2) Supposing that migrations between the patch 1 and the patch 2 are much more rapid than the other processes in the system like growth, maturity and predation. The parameter b is the migration rate from the former to the latter while a is the inverse rate (a þ b 6 1). According to the above assumptions, we can formulate the following predator–prey system incorporating the effect of refuge used by prey explicitly
dx1 x1 cx1 y ¼ ðax2 bx1 Þ þ e r 1 x1 1 d1 x1 a þ x1 ds K1 dx2 ¼ ðbx1 ax2 Þ þ eðd2 x2 Þ ds dy ecx1 ¼e d y ds a þ x1
ð2Þ
where x1 ðtÞ and x2 ðtÞ represent the density of prey population in patch 1 and 2, respectively, yðtÞ denotes the density of predators in patch 1. The parameters r1 ; K 1 ; c; di ði ¼ 1; 2Þ; d; e and a are all positive constants and have its biological meanings accordingly. r 1 is the intrinsic per capita growth rate of prey population in patch 1. K 1 is the common environmental carrying capacity. c is the attack rate of predators to prey population for patch 1. di is the mortality rate of prey population in patch i. d is the mortality rate of predators in patch 1. e is the efficiency with which predators convert conx1 sumed prey into new predators. The term aþx denotes the func1 tional response of the predator population in which a is the half saturation constant. This response function is termed as Holling II functional response. We choose the total densities to represent the system globally, that is, the aggregated variables are
x ¼ x1 þ x2 ;
y ¼ y;
_ xðtÞ ¼ _ yðtÞ ¼
bx
aþb
;
x ; 1 þ ab
aða þ bÞ þ ax
ð4Þ
dy
3. Positivity and boundedness
Theorem 3.1. All the solutions of model (4) which start in R2þ are positive and uniformly bounded. Proof. The positively of the solutions of model (4) can be easily obtained according to their ecological meanings. Next, we will only show the boundedness. Defining the function WðtÞ ¼ xðtÞ þ 1e yðtÞ. Hence, we have
a2 r 1 x d _ ¼ ar 1 bd2 x 1 W y: K 1 ðar 1 bd2 Þða þ bÞ e aþb Now, for each V > 0, we get
_ þ VW 6 K 1 ðar 1 bd2 Þða þ bÞ W 4a2 r 1 ðar 1 bd2 Þ
2
ar1 bd2 2 1 ðd VÞ: Vþ e aþb
Let us choose V > d, then the right hand is positive. Thus, the right hand side is bounded for all ðx; yÞ 2 R2þ . _ þ VW < U. Hence, we choose a U such that W Applying the theory of differential inequality, we obtain that
0 < Wðx; yÞ <
U U 1 eVt þ W ðxð0Þ; yð0ÞÞeVt ! ; V V
t ! þ1:
Therefore, all the solutions of model (4) starting in R2þ are confined to the region D, where
D¼
ðx; yÞ 2 R2þ jW ¼
U þ e; e > 0 : V
h
4. The equilibria
x1 ¼ x x2 ;
The possible equilibria of model (4) can be obtained by solving the following equations
or
x1 ¼
aecxy
In the following sections, we will consider the dynamic behaviors of the above aggregated model.
x is invariant for the fast part. Migration changes the proportions of prey on the two patches, but not their total density x. The solutions of the fast part is described as follows
x2 ¼
ðar1 bd2 Þx a2 r1 x2 acxy aþb K 1 ða þ bÞ2 aða þ bÞ þ ax
x2 ¼ x x1 :
b The parameter aþb represents a constant proportion of prey population staying on patch 2 and denotes the constant proportion of prey using refuge. The parameter ab is the ratio of the migration rate from patch 1 to patch 2 to that from patch 2 to patch 1 and represents the migration efficiency. Therefore, the aggregated model can be obtained by following steps.
aecxy aða þ bÞ þ ax
We can easily calculate all equilibria of model (4) are P0 ð0; 0Þ,
adða þ bÞ ; aðec dÞ
and P2 ðx ; y Þ, where
y ¼
er1 a2 r1 x x 1 : d K 1 ðar1 bd2 Þða þ bÞ
The equilibrium point P2 ðx ; y Þ is positive if and only if
b ð3Þ
(II) Substituting x1 and x2 into model (3), we obtain the aggregated system at the slow time scale
ð5Þ
dy ¼ 0
2 ÞðaþbÞ P1 ðK 1 ðar1 bd ; 0Þ a2 r1
x ¼
(I) Adding the first two equations of model (2), we have
dx1 x1 cx1 y d1 x1 d2 x2 ¼ e r 1 x1 1 a þ x1 dt K1 dy ecx1 ¼e d y ds a þ x1
ðar1 bd2 Þx a2 r1 x2 acxy ¼0 aþb K 1 ða þ bÞ2 aða þ bÞ þ ax
aþb
<1
K 1 d2 ðec dÞ ; K 1 ðr 1 þ d2 Þðec dÞ adr1
or
b
a
<
r1 ad : 1 K 1 ðec dÞ d2
128
Z. Ma et al. / Mathematical Biosciences 243 (2013) 126–130
Obviously, the equilibrium density of prey population x increases as the proportion of prey refuge increases according to b the representation of x with respect to aþb and increases with b the ratio a. On the other hand, one can easily see that y is a continb uous function of parameter aþb , simple computation shows that
dy aer1 ¼ b d aþb ðec dÞ 1
8 2 32 9 b < = 1 aþb adr1 ðr 1 þ d2 Þ 4 5 : 2 :1 b b K 1 ðec dÞ 1 aþb d2 ; r 1 aþb
b aþb
Hence, considering into two cases qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 ðecdÞ 1 adr 1 ðr1 þd2 Þ
b Case 4.1. If 0 < aþb < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K 1 ðecdÞ ðr þd2 Þ1 adr 1 ðr 1 þd2 Þ 1
, then y is a strictly increas-
ing function. That is, increasing the amount of prey refuge can increase the equilibrium density of predators. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 ðecdÞ 1 adr 1 ðr 1 þd2 Þ
Case 4.2. If qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K 1 ðecdÞ ðr þd2 Þ1 adr 1 ðr1 þd2 Þ 1
b < aþb , then y is a strictly decreasing
function. That is, increasing the proportion of prey refuge will decrease the equilibrium density of predators.
Again, we can easily see that y is a continuous function of parameter ab and simple computation obtains that
" # dy aer1 adr1 ðr1 þ d2 Þ ¼ 1 2 : ec d d ab K 1 ðec dÞ r 1 b d2 a
Similarly, we consider into two cases qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þd2 Þ , then y is a strictly increasCase 4.3. If 0 < ab < dr12 ½1 Kadðr 1 r 1 ðecdÞ ing function. That is, increasing the amount of prey refuge can increase the equilibrium density of predators. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þd2 Þ < ab, then y is a strictly decreasing Case 4.4. If dr12 ½1 Kadðr 1 r 1 ðecdÞ function. That is, increasing the proportion of prey refuge will decrease the equilibrium density of predators.
5. Stability analysis
J1 ¼ @
or
b
a
>
r1 ad : 1 K 1 ðec dÞ d2
Otherwise, it is locally unstable and the equilibrium point P2 ðx ; y Þ lies in fðx; yÞjx > 0; y < 0g. Again, the Jacobian matrix for the interior equilibrium point P2 ðx ; y Þ is as follows
J2 ¼
a11
a12
a21
0
where
r1 a2 r1 ðec þ dÞx d ; ec K 1 ðar 1 bd2 Þða þ bÞ aða þ bÞaecy ¼ > 0: ½aða þ bÞ þ ax 2
a11 ¼ a21
a12 ¼
cx < 0; a þ x
Clearly, DetJ 2 ¼ a12 a21 > 0, hence, the sign of the eigenvalues depends only on trJ 2 ¼ a11 . Therefore, the interior equilibrium point P 2 ðx ; y Þ is locally asymptotically stable if and only if a11 < 0, that is
b
aþb
>1
K 1 d2 ðec dÞ ; K 1 ðr 1 þ d2 Þðec dÞ ar1 ðec þ dÞ
or
b
a
>
r1 aðec þ dÞ 1 : K 1 ðec dÞ d2
Otherwise, the interior equilibrium point is locally unstable. Next, we will prove the global stability of the interior equilibrium point. Let us consider model (4) in the following form
_ xðtÞ ¼ xgðxÞ ypðxÞ _ yðtÞ ¼ yðqðxÞ dÞ
ð6Þ
Where
In this section, we will consider the stability properties of the equilibria of model (4). Clearly, the equilibrium point P 0 ð0; 0Þ is a saddle point and is unstable. The Jacobian matrix of model (4) at the predator-free equilib2 ÞðaþbÞ rium point P 1 ðK 1 ðar1 bd ; 0Þ is given by a2 r
0
That is
b K 1 d2 ðec dÞ ; >1 K 1 ðr 1 þ d2 Þðec dÞ adr1 aþb
1
1
r
1 ðar 1 bd2 Þ aarcK1 þK 1 ðar1 bd2 Þ
0
ecK 1 ðar 1 bd2 Þ aar1 þK 1 ðar1 bd2 Þ d
A
2 ÞðaþbÞ Thus, The predator-free equilibrium point P1 ðK 1 ðar1 bd ; 0Þ is a2 r1 locally asymptotically stable if and only if
ecK 1 ðar 1 bd2 Þ d < 0: aar1 þ K 1 ðar1 bd2 Þ
0
p ðxÞ 0 d xg ðxÞ þ gðxÞ xgðxÞ pðxÞ dx qðxÞ d
!
gðxÞ ¼
ar1 bd2 a2 r1 x acx ; pðxÞ ¼ ; qðxÞ ¼ epðxÞ: aða þ bÞ þ ax aþb K 1 ða þ bÞ2
We have the following Lemma regard uniqueness of limit cycles of model (4). Lemma 5.1. Supposing in model (6)
! p0 ðxÞ 0 d xg ðxÞ þ gðxÞ xgðxÞ pðxÞ 60 dx qðxÞ d in 0 6 x < x and x < x 6 K. Then model (6) has exactly one limit cycle which is globally asymptotically stable with respect to the region D ¼ fðx; yÞjx > 0; y > 0g n fE2 ðx ; y Þg.
3 2 ðar 1 bd2 Þa2 r 1 x ar1 bd2 a2 r1 x ar1 bd2 a2 r 1 x aa d 4 K 1 ðar1 bd2 ÞðaþbÞ2 þ aþb 1 K 1 ðar1 bd2 ÞðaþbÞ aþb 1 K 1 ðar1 bd2 ÞðaþbÞ aðaþbÞþax 5 6 0 () aecx dx d aðaþbÞþax " # aðaþbÞ K 1 ðar 1 bd2 ÞðaþbÞ d xð2x þ a a2 r1 6 0 () P 0: adðaþbÞ dx x aðecdÞ
Z. Ma et al. / Mathematical Biosciences 243 (2013) 126–130
According to Lemma 5.1, we obtain that It is equivalent to showing that
2 adða þ bÞ adða þ bÞ K 1 ðar 1 bd2 Þða þ bÞ aða þ bÞ x þ 2a2 r 2a aðec dÞ aðec dÞ 2 adða þ bÞ P 0: aðec dÞ aþbÞ 2 Since ðx adð aðecdÞ Þ > 0 and
adðaþbÞ aðecdÞ
> 0, the above inequality is
equivalent to the following form
adða þ bÞ K 1 ðar 1 bd2 Þða þ bÞ aða þ bÞ : < 2a2 r1 2a aðec dÞ That is
b
aþb or
b
a
<
<1
K 1 d2 ðec dÞ ; K 1 ðr1 þ d2 Þðec dÞ ar 1 ðec þ dÞ
r1 aðec þ dÞ : 1 K 1 ðec dÞ d2
Hence, model (4) has exactly one limit cycle which is globally stable if and only if
b
aþb or
b
a
<
<1
K 1 d2 ðec dÞ ; K 1 ðr1 þ d2 Þðec dÞ ar 1 ðec þ dÞ
r1 aðec þ dÞ 1 : K 1 ðec dÞ d2
Noticing that the nonexistence of limit cycles also implies the globally asymptotical stability of the interior equilibrium point of model (4). Therefore, we have the following main results 1 þaÞ Theorem 5.2. Suppose that e > dðK cðK 1 aÞ , we have
b 1 d2 ðecdÞ (1) If 0 < aþb < 1 K 1 ðr1 þdK2 ÞðecdÞar , then the predator and prey 1 ðecþdÞ
populations periodically oscillate around the unique equilibrium point in the first quadrant. K 1 d2 ðecdÞ b 1 d2 ðecdÞ < aþb < 1 K 1 ðr1 þd , (2) If 1 K 1 ðr1 þd2KÞðecdÞar 1 ðecþdÞ 2 ÞðecdÞadr 1
then
the two interacting populations tend to reach a globally asymptotically stable equilibrium point in the first quadrant. K 1 d2 ðecdÞ b < aþb < 1, then predators go extinct (3) If 1 K 1 ðr1 þd 2 ÞðecdÞadr 1
129
and incorporating the effect of refuge used by prey in the considered models implicitly. However, we present another approach allowing to propose a patchy predator–prey model with one patch as refuge and incorporate prey refuge in the predation model explicitly. The results show that the refuge used by prey plays an important role in interacting populations dynamic consequences and the equilibrium density of two interacting populations. Furthermore, the impact of the migration efficiency on the proposed model is also investigated. The analysis on the stability of the predator–prey system reveals the following conclusions 1 The equilibrium density of prey population increases as refuge used by prey and/or the migration efficiency increasing while that of predators decreases with prey refuge and/or the migration efficiency. Although several works have shown an increase in prey density as the refuge used by prey, the effect on predator density is less clear. Our results are consistent with those of lez-Olivars and Ramos-Jiliberto [13] and contrast with Gonza those of McNair [17] who predicted an increase in the equilibrium density of predator population with prey refuge. 2 When the effect of prey refuge and/or the migration efficiency are strong enough, the predators will extinct and the prey population reach their environmental carrying capacity eventually. This behavior also observed by Collings [16] and Ma et al. [18]. 3 Our results show that the effect of prey refuge and/or the migration efficiency play an important role in determining the stability of the interior equilibrium point of the considered model. The refuge used by prey and/or the migration efficiency can increase the global stability of the interior equilibrium point (stabilizing effect) which is agreement with most previous results on simple models [6,7,9,13–15] and disagreement with that of Ma et al. [18] who found a clear destabilizing effect on the considered model. In this paper, increase in global stability or stabilization refers to cases where an interior equilibrium point changes from repeller to an attractor due to changes in the values of the controlled parameter [See Ma et al. [18]]. Increase in stability of the interior equilibrium point of the considered model with increasing prey refuge is usually associated with predator population which exhibit saturating functional responses to prey population.
Acknowledgements
while prey population reaches its maximum environmental carrying capacity.
1 þaÞ , we obtain that Theorem 5.3. Suppose that e > dðK cðK 1 aÞ
(1) If 0 < ab < dr12 ½1 KaðecþdÞ , then the prey population and preda1 ðecdÞ tors periodically oscillate around the unique equilibrium point in the first quadrant. ad (2) If dr12 ½1 KaðecþdÞ < ab < dr12 ½1 K 1 ðecdÞ , then the two interacting 1 ðecdÞ populations tend to reach a globally asymptotically stable equilibrium point in the first quadrant. ad < ab, then the prey population reaches its max(3) If dr12 ½1 K 1 ðecdÞ imum environmental carrying capacity while predators go extinct. 6. Conclusion When considering the prey refuge in predator–prey systems, one usually concentrates on modifying the functional response
This work was supported by the National Natural Science Foundation of China (No. 11126183) and the Fundamental Research Funds for the Central Universities (No.lzujbky-2011-48). References [1] C.S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol. 91 (1959) 385. [2] M.P. Hassel, R.M. May, Stability in insect host–parasite models, J. Anim. Ecol. 42 (1973) 693. [3] J.M. Smith, Models in Ecology, Cambridge Univ Press, Cambridge, UK, 1974. [4] W.W. Murdoch, A. Stewart-Oaten, Predation and population stability, Adv. Ecol. Res. 9 (1975) 1. [5] M.P. Hassell, The Dynamics of Arthropod Predator–Prey Systems, Princeton Univ. Press, Princeton, NJ, 1978. [6] M.A. Hoy, Almonds (California), in: W. Helle, M.W. Sabelis (Eds.), Spider Mites: Their Biology, Natural Enemies and Control, World Crop Pests, vol. 1B, Elsevier, Amsterdam, 1985. [7] A. Sih, Prey refuges and predator–prey stability, Theor. Popul. Biol. 31 (1987) 1. [8] A.R. Ives, A.P. Dobson, Antipredator behavior and the population dynamics of simple predator–prey systems, Am. Nat. 130 (1987) 431. [9] G.D. Ruxton, Short term refuge use and stability of predator–prey models, Theor. Popul. Biol. 47 (1995) 1. [10] M.E. Hochberg, R.D. Holt, Refuge evolution and the population dynamics of coupled host–parasitoid associations, Evol. Ecol. 9 (1995) 633. [11] J. Michalski, J.C. Poggiale, R. Arditi, P.M. Auger, Macroscopic dynamic effects of migration in patchy predator–prey systems, J. Theor. Biol. 185 (1997) 459.
130
Z. Ma et al. / Mathematical Biosciences 243 (2013) 126–130
[12] V. Kurivan, Effects of optimal antipredator behvior of prey on predator–prey dynamics: the role of reufuges, Theor. Popul. Biol. 53 (1998) 131. [13] E. González-Olivars, R. Ramos-Jiliberto, Dynamics consequences of prey refuges in a simple model system: more prey, few predators and enhanced stability, Ecol. Model. 166 (2003) 135. [14] T.K. Kar, Stability analysis of a prey–predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul. 10 (2005) 681. [15] Y. Huang, F. Chen, L. Zhong, Stability analysis of a prey–predator model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput. 182 (2006) 672.
[16] J.B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator–prey interaction model incorporating a prey refuge, Bull. Math. Biol. 57 (1995) 63. [17] J.M. McNair, The effects of refuges on predator–prey interactions: a reconsideration, Theor. Popul. Biol. 29 (1986) 38. [18] Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang, Z. Li, Effects of prey refuges on a predator–prey model with a class of functional responses: the role of refuges, Math. Biosci. 218 (2009) 73.