Applied Mathematical Sciences, Vol. 8, 214, no. 11, 527-537 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/12988/ams.214.4275 Dynamical Analysis of a Harvested Predator-prey Model with Ratio-dependent Response Function and Prey Refuge N. N. Ilmiyah, Trisilowati* and A. R. Alghofari Department of Mathematics Faculty of Sciences - Brawijaya University Jl. Veteran Malang 65145, Indonesia * Corresponding author Copyright 214 N. N. Ilmiyah, Trisilowati and A. R. Alghofari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper deals with a construction and an analysis of harvested predator-prey model with ratio-dependent response function and prey refuge. The harvesting is applied on both of predator and prey because they have a commercial value, while the prey refuge is applied in accordance with the fact that prey has a refuge instinct enabling to reduce the possibility of prey catching rate. According to the analysis, there are four equilibrium points which are stable under certain conditions, namely the prey extinction, the predator extinction, and two coexistence points. Finally, numerical solutions are presented not only to illustrate each equilibrium points but also to illustrate the effects of prey refuge. Keywords: Dinamical analysis, Harvesting, Predator-prey model, Prey refuge, Ratio-dependent response function. 1 Introduction The study of predator-prey model can be recognized as a major issue in applied mathematics since it was initiated by Lotka and Volterra in the mid 192. For
528 N. N. Ilmiyah, Trisilowati and A. R. Alghofari example, dynamic consequences of prey refuge in a simple model system have been studied in [2]. Similarly, Ji and Wu [3] have studied a predator-prey model with constant rate prey harvesting incorporating a prey refuge. Kar [4] has investigated modeling of predator-prey system with harvesting of predator-prey incorporating a prey refuge. This research used Holling Type II response function with linear harvesting and prey refuge. 11 1, 1 1. Here, and denote the population densities of prey and predator respectively. is the prey intrinsic growth rate. represents the carrying capacity, is a constant number of prey refuge. In model (1),,,, and are positive constants that stand for the death rate of predator, capturing rate, conversion rate and half saturation constant, respectively. and denote the harvesting rates for prey and predator, where, denote the nominal fishing efforts, while and are the catchability coefficients of the prey and predator. Generally, this study used prey-dependent response function which depends on the densities of prey only, see e.g [2], [3], and [4]. Recently, the ratio-dependent model have been observed by Lenzini and Rebaza [5]. This model is appropriate with the study of Arditi and Ginzburg [1] who have seen in accordance with the fact that response function ought to depend on the densities of both predator and prey. The model of Lenzini and Rebaza [5] has followed the scalling of Xiao et.al [8] model. 1 2, where, /, / and /. For simplicity,, and respectively have similar biological meaning with, and, while is a constant harvesting 1
Dynamical analysis of a harvested predator-prey model 529 effort. This paper presents a modification of ratio-dependent response function incorporating with prey refuge. This paper is organised as follows. In Section 2, we construct the model and then the analysis of the existence of the equilibrium points and their stability is given in Section 3. Furthermore, we show some numerical simulations to illustrate the stability of each equilibrium point in Section 4. Finally, a conclusion of the paper is presented in the final section. 2 Mathematical Model The model (2) considers the predator harvesting without incorporating prey refuge. However, the prey refuge should be incorporated in the model, see e.g [3], [4], [7], and [9], according to Pal and Samanta [6], in the fact, refuge instinct of prey is a factor that should be calculated in modeling. The prey will come out of the refuge only when it feels safe from predator. So in this paper, we perform a modification of the model by incorporating a refuge of the prey. Here, we also follow the scaling process of Xiao at. al [8]. So the model in this paper is 1 1 1 3 11 1, where. 3 Equilibrium and Their Stability System (3) has four equilibrium points, namely prey extinction, 1, predator extinction 1,, and two coexistence points, and,, where 1 1 11 and, 4 2
53 N. N. Ilmiyah, Trisilowati and A. R. Alghofari here 1 1 1 2 11 1 1 1 1 1 1 5 6 411 1 1. 7 In the third and the fourth equilibrium points, we assume that, 1. So the harvesting does not lead to the extinction. The existence condition of the prey extinction point is 1, and the existence conditions of the predator extinction point is 1. While The existence conditions of the first coexistence point is one of the following conditions: 1. 1 1, and 8 2. 1 1 and 1 1 9 3. 1 1 and 1 1 11 The existence conditions of the second coexistence point is one of the following conditions: 4. 1 1, and 11 5. 1 1, 1 1 1 and 1 1 The general Jacobian of (3) is 1 2 1 1 1 1 1 1 1 2 1. 1 12 Theorem 1. is asymtotically stable if 1 1, while is asymtotically stable if 1. Proof. The Jacobian at is given by
Dynamical analysis of a harvested predator-prey model 531 1 1 13 1 1 The eigenvalues are 1 and 1 1. So the prey extinction point is asymtotically stable if 1 1. The Jacobian at is 1 14 1 The eigenvalues are 1 and 1. So the prey extinction point is asymtotically stable if 1 Theorem 2. is asymtotically stable in any condition in (9) and (1), and is asymtotically stable if condition (9) is fulfilled. For condition (8) and (11), and is asymtotically stable if and τ. Proof. The Jacobian evaluated at and is where,, 1 2 1 1 1 1 1 1 1 2 1 1. Here,,,and, while surely in the condition (9), (1) and (12). So the determinant and the trace τ are and τ while in the condition (8) and (11), is negative if and τ.
532 N. N. Ilmiyah, Trisilowati and A. R. Alghofari 4 Numerical Simulations In this section, we present simulations of the system (3) using ppline on Matlab. The simulations are divided into two simulations as follows. 1. Simulation I using parameters: 1.8, 1.2,.2,.6, and.4 2. Simulation II using parameters: a) 1.8, 1.2,.2,.66, and.4 b) 1.8, 1.2,.2,.6,.2 and.4 c) 1, 1,.2,.6,.6 and.4 4.1 Simulation I Simulation I illustrates the stability of. Here,,2 is asymtotically stable, while.6, exists but it is unstable. In Figure 1(a), the trajectories are convergent to. It means that if the sum of prey harvesting rate and the catching rate of prey that does not have refuge is greater than the prey growth rate, so the prey will be extinct, while, the predator can survive even though there is no prey to catch. This condition is caused by the predator growth rate that is not only influenced by prey catching rate, but also the other resources. 4.2 Simulation II Simulation II consists of three cases, namely simulation IIa, IIb, and IIc. The simulations illustrate the stability of and which satisfy the existence on condition (8), (9), (1), (11) and (12). 4.2.1 Simulation IIa Simulation IIa illustrates the stability of when 1 1 1 and satisfy the existence on condition (8) and (11). In this analysis,.74,54 and.28,36 are asymtotically stable, while.6, exists but it is unstable. Figure 1(b) shows that using parameters on Simulation IIa, the trajectories are convergent not only to and but also to,2. It indicates that, if the
Dynamical analysis of a harvested predator-prey model 533 sum of prey harvesting rate and the catching rate of prey that does not have refuge is greater than the prey growth rate, so the prey will be extinct, while the predator will exist. But if,,, and are fulfilled concurrently, then both populations can survive. These conditions occur using parameters which satisfies the existence and stability of,, and. 4.2.2 Simulation IIb Simulation IIa illustrates the stability of, this coexistence point satisfies the existence on condition (9). Here, we get.3728,.2176 which is asymtotically stable. While the prey extinction point,2 and the predator extinction point.8, exist but they are unstable. Figure 1(c) shows that the trajectories are convergent to. It indicates that if the sum of prey harvesting rate and the catching rate of prey that does not have refuge is smaller than the prey intrinsic growth rate, then both populations can survive. 4.2.3 Simulation IIc Simulation IIc illustrates the stability of which is agree with the existence on condition (1)..4,42 is asymtotically stable, while,2 and.4, exist but they are unstable. Figure 1(d) shows that the trajectories are convergent to the coexistence point. It is appropriate with the analytic results that state is asymptotically stable. It shows that when the sum of the prey harvesting rate and the catching rate of prey which does not have refuge is the same as the prey intrinsic growth rate, then both populations can survive with the following condition 1 1 1
534 N. N. Ilmiyah, Trisilowati and A. R. Alghofari.25.2 8.2 6 4 y 5 y 2.8.6.5.4.2.2.3.4.5.6.7 x (a).2.3.4.5.6 x (b).3 8 6.25 4.2 2 y 5 y.8.6.5.4.2.2.3.4.5.6.7.8 x (c).5 5.2.25.3.35.4 x (d) Figure 1. The phase portrait of the simulation: (a) simulation I (b) simulation IIa (c) simulation IIb (d) simulation IIc 4.3 The Effect of Prey Refuge We show four cases to illustrate the effect of prey refuge. The first case is a simulation without considering the prey refuge. The second, third and fourth cases are simulations with considering the prey refuge, that are.65,.66 and.7 respectively and the value of the other parameters are 1.8, 1.2,.2, and.4.
Dynamical analysis of a harvested predator-prey model 535 6 4 x(t) y(t) 8 6 x(t) y(t) 2 4 2 populasi.8 populasi.8.6.6.4.4.2.2 5 1 15 2 25 t (a) 5 1 15 2 25 3 t (b).65 7 6 x(t) y(t) 94 92 x(t) y(t) 5 9 populasi 4 3 2 1 populasi 88 86 84 82.9 8.8 78 5 1 15 2 25 3 35 4 t (c).66 76 1 2 3 4 5 6 7 t (d).7 Figure 2. Numerical solutions of system (3) with and without prey refuge Figure 2 illustrates the effect of prey refuge. On Figure 2(a) and 2(b), we can see that prey is convergent to extinction, while predator survives. Figure 2(a) shows that in the absence of prey refuge, prey will be extinct while predator survives. This is due to the harvesting and the prey catching rate is large. We can
536 N. N. Ilmiyah, Trisilowati and A. R. Alghofari see on Figure 2(b) that the presence of prey refuge of.65 can not save the prey from extinction. The refuge only make the prey extinction rate is slower than before. This condition is due to the catching rate is large, so the sum of prey harvesting rate and the catching rate of prey that does not have refuge still exceeds the prey intrinsic growth rate. Another effect of refuge instinct of prey is shown by Figure 2(c) and 2(d). They appear that with the refuge instinct rate.66 and.7, both populations can survive. This is because of the condition (3) for existence and stability of are fulfilled. 5 Conclusions In this work, we have studied the predator prey model with ratio-dependent response function, incorporating prey refuge, and harvesting on both prey and predator. Existence and stability of equilibrium points were investigated analytically and numerically. The numerical simulations with and without incorporating prey refuge show that the incorporating prey refuge on the system enable to reduce the possibility of prey extinction. References [1] R. Arditi, and L. R. Ginzburg, Coupling in Predator-prey Dynamics: Ratiodependence, The Journal of Theoretical Biology, 139 (1989), 311-326. [2] E. Gonzalez-Olivares, and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, 166 (23), 135-146. [3] L. Ji, and C. Wu, Qualitative Analysis of a Predator-prey Model with Constant-rate prey Harvesting Incorporating a Constant Prey Refuge. Nonlinear Analysis: Real World Applications, doi: 116/j.nonrwa.29.7.3.
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