Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting. Peng Feng

Transcription

1 Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting Peng Feng Journal of Applied Mathematics and Computing ISSN J. Appl. Math. Comput. DOI /s z 1 23

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3 J Appl Math Comput DOI /s z ORIGINAL RESEARCH JAMC Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting Peng Feng Received: 4 April 2013 Korean Society for Computational and Applied Mathematics 2013 Abstract In this paper, we investigate the dynamics of a ratio dependent predatorprey model with quadratic harvesting. We examine the existence of the positive equilibria, the related dynamical behaviors of the model, as well as the boundedness and permanence property of the system. We also study the global stability of the interior equilibrium without time delay. Finally some bifurcation analysis is carried out for the system with delay and the results are illustrated numerically. Keywords Time delay Ratio dependent Permanence Hopf bifurcation Mathematics Subject Classification 34C05 34C25 92A15 1 Introduction In 1995, it was reported that at McGrath area of Alaska, moose numbers had declined greatly since the 1970s and wolves were keeping the moose population from increasing. The Board of Game approved a control program to remove 80 percent of the wolves in the area and the moose population was able to come back to support the harvest demand. Indeed, in population management, predator control is frequently employed to achieve sustained yield in the preydeer, antelope, game birds, etc.. For example, the control of foxes, raccoons and skunks can be used to protect duck species in danger of extinction and also the endangered shorebirds. There have been many predator-prey models without predator control since the original Lotka-Volterra model 1926; such as Holling Tanner Type II, III 1959; Leslie model 1945; Rosenzweig-MacArthur model 1963; and Yodzis model P. Feng Department of Mathematics, Florida Gulf Coast University, Fort Myers, FL 33965, USA pfeng@fgcu.edu

4 P. Feng Many of these models have a functional response depending on prey density and their properties are very well understood. In the early 90 s, several biologists proposed that the functional response should depend on the ratio of the prey and predator. Such functional response is called a ratio-dependent response and a typical model takes the following form, x t = xa bx cxy my + x, y = y d + fx. my + x Here, xt and yt represent the population of prey and predator at time t, respectively; a/b is the carrying capacity of the prey, d>0isthe predator death rate. a, c, m and f are positive constants representing the prey growth rate, capturing rate, half saturation constant and the predator conversion rate, respectively. The ratio dependent model has received both support and criticism. One question underlying the debate is how predator population influences the predator s consumption rate, that debate has persisted for more than two decades. However, we will not discuss here the ecological significance of this type of model but rather some general mathematical features of the associated model. For the interested readers, we refer to Arditi [2], Abrams [1] and the references therein. We shall also point out that there are some qualitative analyses of food chains or multi-species interaction models based on ratio-dependent approach, see for example Bake [3] and Liu [6]. It is now well known that the ratio-dependent models are capable of producing richer and more reasonable dynamics. This has been well documented in the study by Kuang and Beretta [5] and by Hsu and Huang [4]. In [5], it was shown that the ratiodependent type models do not produce the so-called paradox of enrichment which states that enriching a predator-prey system will cause an increase in the equilibrium density of the predator but not in that of the prey and will destabilize the community equilibrium. Another characteristic behavior of predator-prey systems is the oscillatory phenomenon of the predator and prey densities. Another mechanism that might cause this oscillatory behavior is the introduction of time delays in the models. Indeed, the model we shall study in this paper takes both the delay and predator control into consideration, and it takes the following form, { x t = x1 x αxy x+y, y t = y δ + βxt τ xt τ+yt τ 1 μy2 with the following initial conditions: x 0 θ = φ 1 θ 0, y 0 θ = φ 2 θ 0, θ [ τ,0], x0>0, y0>0, 2 where φ 1,φ 2 C[ τ,0],r 2 +, R2 + ={x, y : x 0,y 0}, and x tθ = xt +θ. In this model, the time delay represents the time that takes for the predator to consume prey and reproduce their next generation. The quantity μy 2 represents the quadratic harvesting effort in the predator. The dynamics of the system without time delay has been treated in Saleh [8]. The system exhibits interesting dynamics around the coexistence equilibria, including multiple bifurcation, periodic solution and homoclinic orbit. Similar models incorporating time delay were treated in numerous articles. In Nindjin [7], the authors

5 Analysis of a delayed predator-prey model with ratio-dependent studied the global stability of interior equilibria of a modified Leslie-Gower model with Holling type-ii scheme. In Xiao [9, 10], the authors considered the non-delayed system with constant harvesting rate. The objective of this paper is to perform a qualitative analysis on this delayed ratio-dependent system with harvesting term. We establish some local stability results for the positive interior equilibria. We also investigate the effect of the delay on the stability of the equilibria. Finally we address the effect of the harvesting term on maintaining a sustainable prey size. The paper is organized as follows. In Sect. 2, we present some boundedness and permanence results. In Sect. 3, we carry out some detailed analysis on the global stability of 1, 0 and also the local stability of interior equilibria without delay. Section 3 also provides some Hopf bifurcation analysis on the system with delay. In Sect. 4, numerical simulations are used to illustrate some of our results. 2 Boundedness and permanence In this section, we present some preliminary results including the boundedness of solutions, permanence, and global stability of 1, 0. We begin first with the existence of interior equilibrium solutions. These equilibrium solutions are determined analytically by setting x t = y t = 0. They are independent of the delay τ. It is easy to verify that this system has two boundary equilibrium points, O = 0, 0, A = 1, 0. The interior equilibrium points are given by E 1 = x 1,y 1, E 2 = x 2,y 2, where x 1,2 = 2β + αδ + μ 2β± α δ μ 2 + 4μβ + αδ αβ, 2β + αμ y 1,2 = x 1,21 x 1,2 x 1,2 + α 1. Thus, we have the following lemma about the existence of interior equilibria. Lemma 1 a System 1 has no equilibrium in the first quadrant if δ μ 2 + 4μβ + αδ αβ < 0. b System 1 has two equilibriums in the first quadrant if δ μ 2 + 4μβ + αδ αβ > 0, α>1, and 0 <x i < 1, i = 1, 2. Lemma 2 The positive quadrant is invariant for system 1. Proof To prove that for all t [0, AA > 0, xt > 0 and yt > 0 under the initial conditions x0>0, y0>0, we suppose otherwise that there exists a 0 <T <A such that for all t [0,T, xt > 0, yt > 0 and either xt = 0oryT = 0. For any t [ τ,t, we obtain the following integral equation from 1 { t xt = x0 exp αys 0 1 xs xs+ys ds, yt = y0 exp t βxs τ 0 δ μys + xs τ+ys τ ds. 3

6 P. Feng Since xt and yt are both continuous on [ τ,t, there exists a positive constant M such that for all t [ τ,t, { t xt = x0 exp αys 0 1 xs xs+ys ds x0e TM, yt = y0 exp t 0 δ μys + βxs τ xs τ+ys tau ds 4 y0e TM. Taking t T, we get xt > 0 and yt > 0, a contradiction. Thus xt > 0, yt > 0 for any t [0,A. Lemma 3 Let xt, yt be the solution of the system 1. Then lim sup xt 1, t + and provided that β>δ. lim sup yt β/δ 1e βτ t + Proof From the first equation of system 1, we have that for all t [0, x t xt 1 xt. Let xt be the solution of the following initial value problem { x t = xt1 xt, x0 = x0>0. 5 By standard comparison principle, we have xt xt for all t [0, +. Thus lim sup xt lim sup xt = 1. t + t + From the second equation, we have y t < βyt. 6 Thus yt y0e βt. 7 Thus, for t>τ, integrating 6on[t τ,t], we obtain yt τ yte βτ. Note that there exists T>0such that for t>t, xt < 1. Hence for t>t+ τ, y β β t yt δ μyt yt 1 + ye βτ 1 + yte βτ δ. 8

7 Analysis of a delayed predator-prey model with ratio-dependent Apply a similar argument we obtain that lim sup yt < β/δ 1e βτ. t + Hence if β>δ, the system is bounded. Definition 1 System 1 is called permanent if there exist m, M, 0<m<M, such that for all solutions of 1, { } min lim inf xt,lim inf yt m, t + t + { } max lim sup xt,lim sup yt M. t + t + Theorem 1 The system 1 is permanent if α<1 and β>δ+ μm where M = max{1,β/δ 1e βτ }. Proof Let so that { max M = max { 1,β/δ 1e βτ}, lim sup t + From the prey equation of system 1, we have } xt,lim sup yt M. t + x t > xt 1 xt α. Hence lim inf xt 1 α>0. t + For any ν>1, there exists a positive T ν such that for t>t ν, xt > 1 α/ν and yt < νm. Then for t>t ν + τ, β 1 α y ν t yt 1 α ν + νm δ μνm This leads to y t > δ μνmyt, which involves, for t>t ν + τ, Thus for t>t ν + τ, y t yt. yt τ < yte δ+μνmτ. 9 1 α ν β 1 α ν δ μνm + yte δ+μνmτ,

8 P. Feng which yields β 1 α lim inf yt ν t + δ + μm 1 α e δ+μνmτ. ν Taking ν 1, we get β δ μm1 α lim inf yt e δ+μmτ := y 1. t + δ + μm Let m = min{y 1, 1 α} > 0. Then we have shown that system 1 is permanent. Definition 2 System 1 is said to be not persistent if min lim inf xt,lim inf yt = 0. t + t + Theorem 2 If α>1 + δ, the system is not persistent. Proof If α>1 + δ, then there exists an ɛ>0such that α 1 + ɛ = 1 + δ. We let x0 xt y0 <ɛand we claim that for all t>0, yt <ɛ. Otherwise, there exists t 0 > 0 such that Then for t [0,t 0,wehave from which we obtain Similarly, from which we obtain Thus for t [0,t 0 ], xt 0 yt 0 = ɛ and for t [0,t 0, x t xt 1 α, 1 + ɛ xt x0e 1 α 1+ɛ t = x0e δt. y t yt δ + μyt, yt y0e δt. α 1+ɛ t xt yt x0e1 y0e δt xt yt <ɛ. = x0 y0 <ɛ. This in turn implies that xt x0e δt for all t 0. That is lim t + xt = 0. Hence the system is not persistent if α>1 + δ.

9 Analysis of a delayed predator-prey model with ratio-dependent Theorem 3 If α>1 + δ, β< such that α 1 δ αδ, then there exist positive solutions xt, yt lim xt,yt = 0, 0. t + Proof When α>1 + δ, we have lim t + xt = 0 and for t 0, xt yt α 1 + δ 1, provided that Hence for t τ, which implies x0 y0 α 1 + δ 1. y β t yt 1 + α 1 δ 1+δ δ μy, αδ lim yt = 0, if β< t + α 1 δ. 3 Stability analysis 3.1 Global stability of 1, 0 Theorem 4 If β<δ, α<1, then 1, 0 is globally asymptotically stable. Proof If β<δ, clearly lim t + yt = 0 and lim inf t + xt 1 α. Thus for any 0 <ɛ<1, there exists Tɛsuch that for t>tɛ, xt 1 xt ɛ x t xt 1 xt. Hence lim t + xt = Local stability of E without delay Theorem 5 If α<1, β>δ+μm with M = max{1,β/δ 1} and α> 1 x 2m where m is as defined in Theorem 1. Then the interior equilibrium E is locally asymptotically stable. Proof We let Vx,y = x x x ln x/x + c y y y ln y/y, where c is a constant to be chosen later.

10 P. Feng Vx,y is continuous on the interior of the positive quadrant. It is also easy to verify that V is positive for all values of x and y except for the positive interior equilibrium E where it is zero. The time derivative of V along a solution of system 1 is dv dt = x x 1 x αy + c y y δ μy + βx x + y x + y = x x x x + αy x x αx y y x + y x + y + c y y μ y y + βy x x βx y y x + yx + y = c μ + αy x x 2 x + y x + y βx x + yx + y y y 2 αx x + y x + y + cβy x + y x + y x x y y. Hence if we pick c = αx βy, and if α<1, β>δ+ μm with M = max{1,β/δ 1}, for t>t,wehavext,yt > m where m is defined as in Theorem 1. Hence dv dt 1 + αy 2mx + y x x 2 + c μ βx 2mx + y y y 2. We can then easily verify that dv/dt is negative definite if α> 1 x 2m. Thus E is locally asymptotically stable. 3.3 Hopf bifurcation We shall consider the linearized system at the positive interior equilibria. { u t = a 11 ut + a 12 vt, v t = a 21 ut τ+ a 22 vt τ μy vt, 10 where a 11 = x + αx y x + y 2 = x + 1 x x + α 1, α a 12 = αx 2 + α 1 2 x + y 2 = x, α a 21 = a 22 = βy 2 x + y 2 = β1 x 2 α 2, βx y x + y 2 = β1 x x + α 1 α 2.

11 Analysis of a delayed predator-prey model with ratio-dependent The characteristic equation is where λ 2 + Aλ + B + Cλ + De λτ = 0, 11 A = a 11 + μy, B = a 11 μy, C = a 22, D= a 11 a 22 a 12 a 21. Let λ = iω ω>0 be a root of 11. Then we have From this it follows that Cωsin τω+ D cos τω= ω 2 B, Cωcos τω D sin τω= Aω. ω 4 + A 2 C 2 2B ω 2 + B 2 D 2 = If B 2 D 2 < 0 holds, then 12 has a unique positive root ω 0, ω 0 = A 2 C 2 2B+ A 2 C 2 2B 2 4B 2 D 2. 2 The corresponding critical time delay is τ 0 = 1 ω 0 arccos D ACω2 0 BD C 2 ω D2 + 2nπ ω 0, n= 0, 1, 2,... Differentiating 11 with respect to τ and substituting τ = τ 0, we get { dλ Re dτ } 1 τ=τ 0 = 2ω2 0 + A2 C2 2B A 2 ω ω2 0 B2 = We have the following results. A 2 C 2 2B 2 4B 2 D 2 A 2 ω ω2 0 B2. Theorem 6 If A + C<0,B + D>0 and B D<0, then the positive equilibrium of 1 is asymptotically stable for τ [0,τ 0 and unstable when τ>τ 0. The system undergoes a Hopf bifurcation when τ = τ Numerical simulations In this section, we present some numerical simulations to illustrate our theoretical analysis. In Fig. 1, wetakeα = 1.05, β = 0.566, δ = 0.4, μ = 0.05 with time delay τ = 5. We observe that in this case the solution tends to the equilibrium solution E , InFig.2, we take the same set of parameters except we increases the time delay to τ = 15. In this case, we observe sustained oscilla-

12 P. Feng tions. The phase portrait also shows that a limit cycle appears. In Fig. 3, wetake α = 1.05, β = 0.566, δ = 0.4, μ = 0.25 and τ = 15. In this case, we observe damped oscillations and the phase portrait shows the solution tends to the equilibrium point E , InFig.4, wetakeα = 1.55, β = 0.566, δ = 0.4, μ = 0.05 and τ = 15. In this case, we observe that the populations of both species are driven to extinct and the phase portrait shows that the solution tends to the origin. This numerical simulation illustrates the result in Theorem 3 since the parameters satisfy the conditions in Theorem 3. InFig.5, wetakeα = 0.95, β = 0.36, δ = 0.4, μ = 0.05 and τ = 15. In this case, since α<1 and β<δ, according to Theorem 4, 1, 0 is global asymptotically stable. The phase portrait also shows the solution tends to the boundary equilibrium A = 1, 0. Finally, in Fig. 6, wetakeα = 0.95, β = 0.566, δ = 0.4, μ = 0.05 and τ = 15. This set of parameters satisfies α<1butβ>δ.we still observe sustained oscillations in this case. Fig. 1 τ = 5 Numerical solution a and phase portrait b with α = 1.05, β = 0.566, δ = 0.4, μ = 0.05 and Fig. 2 a Sustained oscillations generated with α = 1.05, β = 0.566, δ = 0.4, μ = 0.05 and τ = 15; b evolution towards a limit cycle corresponding to the sustained oscillations in a

13 Analysis of a delayed predator-prey model with ratio-dependent Fig. 3 a Damped oscillations generated with α = 1.05, β = 0.566, δ = 0.4, μ = 0.25 and τ = 15; b evolution towards an interior equilibrium point corresponding to the damped oscillations in a Fig. 4 a Extinction of both prey and predator with α = 1.55, β = 0.566, δ = 0.4, μ = 0.05 and τ = 15; b evolution towards the origin corresponding to the numerical solution in a Fig. 5 a The prey approaches the carrying capacity while the predator is driven to extinction with α = 0.95, β = 0.36, δ = 0.4, μ = 0.05 and τ = 15; b phase portrait also shows the solution tends to the boundary equilibrium A

14 P. Feng Fig. 6 a Sustained oscillations generated with α = 0.95, β = 0.566, δ = 0.4, μ = 0.05 and τ = 15; b evolution towards a limit cycle corresponding to the numerical solution in a References 1. Abrams, P.: The fallacies of ratio-dependent predation. Ecology 756, Arditi, R., Ginzburg, L.R.: Coupling in predator-prey dynamics: ratio dependence. J. Theor. Biol. 139, Baek, S., Ko, W., Ahn, I.: Coexistence of a one-prey two-predators model with ratio-dependent functional responses. Appl. Comput. Math. 219, Hsu, S.B., Huang, T.W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55, Kuang, Y., Beretta, E.: Global analysis of Gause-type ratio-dependent predator-prey systems. J. Math. Biol. 36, Liu, Z., Zhong, S., Yin, C., Chen, W.: On the dynamics of an impulsive reaction-diffusion predatorprey system with ratio-dependent functional response. Acta Appl. Math. 115, Nindjin, A.F., Aziz-Alaoui, M.A., Cadivel, M.: Analysis of a predator-prey models with modified Leslie-Gower and Holling-type II schemes with time delay. Nonlinear Anal., Real World Appl. 7, Saleh, K.: A ratio-dependent predator-prey system with quadratic predator harvesting. Asian Trans. Basic Appl. Sci. 024, Xiao, D., Jennings, L.: Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting. SIAM J. Appl. Math. 65, Xiao, D., Li, W., Han, M.: Dynamics in a ratio-dependent predator-prey model with predator harvesting. J. Math. Anal. Appl. 3241,