Hindawi Publishing Corporation ISRN Biotechnology Volume 2013, Article ID 898039, 8 pages http://dx.doi.org/10.5402/2013/898039
Research Article Global Stability of Predator-Prey System with Alternative Prey Banshidhar Sahoo Department of Mathematics, Daharpur A.P.K.B Vidyabhaban, Paschim Medinipur, West Bengal, India Correspondence should be addressed to Banshidhar Sahoo; [email protected] Received 1 April 2012; Accepted 7 June 2012 Academic Editors: M. Tamaoki and E. Villegas Copyright © 2013 Banshidhar Sahoo. is 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. A predator-prey model in presence of alternative prey is proposed. Existence and local stability conditions for interior equilibrium points are derived. Global stability conditions for interior equilibrium points are also found. Bifurcation analysis is done with respect to predator’s searching rate and handling time. Bifurcation analysis con�rms the existence of global stability in presence of alternative prey.
1. Introduction e classical predator-prey model based on the logistic growth principle and Hollings predation theory is as follows: 𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 𝑥𝑥 = 𝑥𝑥 1 − − , 𝑑𝑑𝑑𝑑 𝑘𝑘 1 + 𝑐𝑐𝑐𝑐𝑐
(1) 𝑏𝑏𝑏𝑏𝑏𝑏 𝑑𝑑𝑑𝑑 − 𝑑𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 1 + 𝑐𝑐𝑐𝑐𝑐 where 𝑥𝑥 and 𝑦𝑦 represent the density of prey and predator species with carrying capacity 𝑘𝑘. e constant 𝑎𝑎 denotes the food intake rate of predator, 𝑏𝑏 denotes the food conversion rate to predator, and 𝑑𝑑 is the predator’s death rate. e constant 𝑐𝑐, ℎ being predator’s searching rate, and handling time on 𝑥𝑥, respectively. In this model, there is no protection for prey from predator and predator’s survival depends on prey alone. Here the predator species 𝑦𝑦 totally depends on the prey species 𝑥𝑥 and so there is high predation pressure on the prey species. As a result, the prey species has high extinction risk for different searching rate and handling time which is shown in Figure 1. In nature, when the prey population falls below a certain level, the predator searches alternative prey and returns only when the prey population rises to required level. ere are large numbers of three or more species food chain system [1, 2] instead of two species system for the survival of prey species. Van Baalen et al. [3] showed the switching fashion from prey species to alternative prey for persistence of predator-prey system. Plants bene�t from providing food to predators even when
it is also edible to herbivores which is discussed by Van Rijn et al. [4]. Harwood and Obrycki [5] investigated the role of alternative prey in sustaining predator populations. e role of alternative food for biological pest control in predator-prey system is investigated by many scientists [6, 7]. Sahoo [8] studied a food chain model with different functional responses and different growth rates in presence of additional food for construction of real food chain model. Recently, Sahoo [9] showed that additional food is very important for survival of consumer species in an ecosystem. e consequences of providing a predator with additional food and the corresponding effects on the predator-prey dynamics with monotonic and nonmonotonic functional response and its utility in biological control is comparatively studied by Sahoo [10]. But, all of them assumed that the additional food is not dynamic but maintained at a speci�c constant level either by the nature or by an external agency. In this context, I have proposed a predator-prey model with alternative prey (a dynamic additional food for predator). is model is similar to two prey one predator model. e following assumptions are done to formulate the model. (a) Let 𝑥𝑥 be the prey density, let 𝑦𝑦 be the density of alternative prey and the density of the predator is 𝑧𝑧.
(b) Both preys are distributed uniformly in the habitat.
(c) e prey and alternative prey grow as per logistic equation in the absence of predators.
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Predator
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Population density
c = 0.8
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Handling time h
Handling time h
F 1: Bifurcation diagram of the prey and predator species with respect to handling time ℎ for 𝑘𝑘 𝑘 𝑘, 𝑎𝑎 𝑎 𝑎𝑎𝑎, 𝑏𝑏 𝑏𝑏𝑏𝑏, and 𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑.
(d) e predator-prey and predator-alternative prey capture rates are of Holling type II. (e) e constant 𝑎𝑎 is predator’s handling time on 𝑥𝑥 and 𝑐𝑐, ℎ are predator’s searching rate and handling time on 𝑦𝑦, respectively.
With the above assumptions, we formulate the following model as 𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑 𝑥𝑥 = 𝑥𝑥 1 − − , 𝑑𝑑𝑑𝑑 𝑘𝑘1 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
𝑦𝑦 𝑞𝑞𝑞𝑞𝑞𝑞 𝑑𝑑𝑑𝑑 = 𝑦𝑦 1 − − , 𝑑𝑑𝑑𝑑 𝑘𝑘2 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 𝜖𝜖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧 = − 𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
dissipativeness of the system. e local stability and global stability of the interior equilibrium points of the system are examined in Section 3. Moreover, we discuss the numerical experiment of our system in Section 4. Finally, conclusion is written in Section 5.
2. Theoretical Analysis 2.1. Positive Invariance. Let 𝑋𝑋 𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑇𝑇 ∈ 𝑅𝑅3 and
(2)
where 𝑘𝑘1 and 𝑘𝑘2 are the carrying capacity of prey (𝑥𝑥) and alternative prey (𝑦𝑦), respectively; the constants 𝑝𝑝 and 𝑞𝑞 are predator’s (𝑧𝑧) food intake rate on prey and alternative prey respectively. e constants 𝜖𝜖 and 𝑐𝑐𝑐𝑐 are conversion rates of prey and alternative food to predator, respectively; 𝑑𝑑 is constant death rate for predator. Here we assume that predator’s food intake rate on prey (𝑥𝑥) is much more greater than that of alternative prey (i.e, 𝑝𝑝 𝑝 𝑝𝑝). e parameters 𝑐𝑐 and ℎ characterize the alternative prey. is formulation implies that the density of prey (𝑥𝑥) and alternative prey (𝑦𝑦) are scaled with respect to search rate of the predators, this can be done without loss of generality. e system has to be analyzed with the following initial conditions: 𝑥𝑥𝑥𝑥𝑥 𝑥 𝑥, 𝑦𝑦𝑦𝑦𝑦𝑦𝑦, 𝑧𝑧𝑧𝑧𝑧𝑧𝑧. e main objective of this paper is to investigate the dynamic properties and behaviors of the system. Here I shall analyze the dynamics of the system with respect to predator’s (𝑧𝑧) searching rate 𝑐𝑐 and handling time (ℎ) on alternative prey 𝑦𝑦. is paper is organized as follows. In Section 2, we show the
𝑥𝑥 1 −
𝑝𝑝𝑝𝑝𝑝𝑝 𝑥𝑥 − 𝑘𝑘1 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
𝐹𝐹1 (𝑋𝑋) 𝑦𝑦 𝑞𝑞𝑞𝑞𝑞𝑞 𝑦𝑦 1 − − 𝐹𝐹 (𝑋𝑋) = 𝐹𝐹2 (𝑋𝑋) = , 𝑘𝑘2 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐹𝐹3 (𝑋𝑋) 𝜖𝜖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧 − 𝑑𝑑𝑑𝑑 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (3)
3 where 𝐹𝐹𝐹𝐹𝐹𝐹 𝐹 𝐹𝐹+ → 𝑅𝑅3 and 𝐹𝐹 𝐹 𝐹𝐹∞ + (𝑅𝑅 ). en system (2) becomes
𝑋𝑋̇ 𝑋𝑋𝑋 (𝑋𝑋)
(4)
with 𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋0 ∈ 𝑅𝑅3+ . It is easy to verify that whenever choosing 𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋3 such that 𝑋𝑋𝑖𝑖 =0 then [𝐹𝐹𝑖𝑖 (𝑋𝑋𝑋𝑋𝑋𝑋𝑖𝑖 =0 ≥0 (for 𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖). Now any solution of (4) with 𝑋𝑋0 ∈ 𝑅𝑅3+ , say 𝑋𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋0 ), is such that 𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋3+ for all 𝑡𝑡 𝑡 𝑡 (Nagumo, [11]). eorem 1. All the solutions of the system (2) which initiate in 𝑅𝑅3+ are uniformly bounded.
Proof. Let (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 be any solution of the system (2) with positive initial conditions.
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Let us consider that 𝑤𝑤 𝑤 𝑤𝑤 𝑤 𝑤𝑤 𝑤 𝑤𝑤𝑤
that is,
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = + + . 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
erefore,
𝑦𝑦 𝑝𝑝𝑝𝑝𝑝𝑝 𝑥𝑥 𝑑𝑑𝑑𝑑 =𝑥𝑥 1 − − +𝑦𝑦 1 − 𝑑𝑑𝑑𝑑 𝑘𝑘1 1 + 𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎 𝑘𝑘2 erefore,
that is,
𝜖𝜖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧 𝑞𝑞𝑞𝑞𝑞𝑞 − + − 𝑑𝑑𝑑𝑑𝑑 1 + 𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎 1 + 𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 ≤ 𝑥𝑥 (1 − 𝑥𝑥) +𝑦𝑦 1 − 𝑦𝑦 − 𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ≤ 2 − 𝜃𝜃 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 , 𝑑𝑑𝑑𝑑
where 𝜃𝜃 𝜃 𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃:
𝑑𝑑𝑑𝑑 + 𝜃𝜃𝜃𝜃𝜃𝜃𝜃 𝑑𝑑𝑑𝑑
(5)
e Jacobian matrix of the system (2) at the interior equilibrium point 𝐸𝐸∗ is
(6) where
𝐴𝐴11 = 1 −
(7)
𝐴𝐴12 =
𝐴𝐴13 =
(8)
𝐴𝐴21 =
𝐴𝐴23 =
(10)
𝐴𝐴31 =
0 < 𝑤𝑤 𝑤
+ 𝑤𝑤 𝑥𝑥 (0) ,𝑦𝑦 (0) ,𝑧𝑧 (0) 𝑒𝑒−𝜃𝜃𝜃𝜃 .
𝜃𝜃
𝐴𝐴32 =
(11)
(13)
𝑝𝑝 1 +𝑐𝑐𝑐𝑐𝑐∗ 𝑧𝑧 2𝑥𝑥∗ − , 2 𝑘𝑘1 1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝∗ 𝑧𝑧∗
2
,
𝑞𝑞𝑞𝑞𝑞𝑞∗ 𝑧𝑧∗
2
,
1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗ −𝑝𝑝𝑝𝑝∗ , 1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
𝐴𝐴22 = 1 −
(9)
Applying the theory of differential inequality we obtain 2 1 − 𝑒𝑒−𝜃𝜃𝜃𝜃
𝐴𝐴11 𝐴𝐴12 𝐴𝐴13 𝐽𝐽 𝐸𝐸∗ = 𝐴𝐴21 𝐴𝐴22 𝐴𝐴23 , 𝐴𝐴31 𝐴𝐴32 𝐴𝐴33
𝑞𝑞 1 + 𝑎𝑎𝑎𝑎∗ 𝑧𝑧 2𝑦𝑦∗ − , 2 𝑘𝑘2 1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
−𝑞𝑞𝑞𝑞∗ , 1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
𝜖𝜖 𝑝𝑝𝑝𝑝∗ +𝑐𝑐 ℎ𝑝𝑝 𝑝𝑝𝑝𝑝𝑝 𝑦𝑦∗ 𝑧𝑧∗ 2
1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
𝜖𝜖𝜖𝜖 𝑞𝑞𝑞𝑞∗ + 𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞 𝑥𝑥∗ 𝑧𝑧∗
𝐴𝐴33 = 0.
2
1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
,
(14)
,
For 𝑡𝑡 𝑡 𝑡, we have 0 < 𝑤𝑤 𝑤𝑤𝑤𝑤𝑤. Hence all the solutions of the system (2) that initiate in 𝑅𝑅3+ are con�ned in the region 𝑆𝑆 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑆 𝑆𝑆3+ ∶ 𝑤𝑤 𝑤 𝑤𝑤𝑤𝑤 𝑤 𝑤𝑤, for any 𝜂𝜂 𝜂 𝜂𝜂, which means that all species are uniformly bounded for any initial value in 𝑅𝑅3+ . is proves the theorem.
e characteristic equation of the Jacobian matrix 𝐸𝐸∗ is given by
3. Stability Analysis
where
3.1. Existence and Local Stability of Interior Equilibrium Points. e interior equilibrium point of the system is given by 𝐸𝐸∗ (𝑥𝑥∗ ,𝑦𝑦∗ ,𝑧𝑧∗ ), where, 𝑦𝑦∗ = (𝑘𝑘1 𝑘𝑘2 (𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝2 𝑞𝑞𝑞𝑞∗ )/𝑘𝑘1 𝑝𝑝, 𝑧𝑧∗ = (𝜖𝜖𝜖𝜖𝜖1 −𝑥𝑥𝑥𝑥𝑥𝑥1 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝2 𝑘𝑘1 +𝑐𝑐𝑐𝑐2 𝑘𝑘2 )𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥1 𝑘𝑘2 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝1 𝑝𝑝𝑝, and 𝑥𝑥∗ is the positive root of the equation 2
𝐴𝐴𝐴𝐴∗ + 𝐵𝐵𝐵𝐵∗ + 𝐶𝐶 𝐶𝐶𝐶
(12)
where, 𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴2 𝑘𝑘1 +𝑐𝑐𝑐𝑐2 𝑘𝑘2 ) − 𝑎𝑎𝑎𝑎1 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝2 𝑞𝑞𝑞𝑞, 𝐵𝐵 𝐵𝐵𝐵𝐵𝐵𝐵21 𝑝𝑝 𝑝 𝑘𝑘1 𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝1 𝑘𝑘2 (𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝1 𝑘𝑘2 𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞1 (𝑝𝑝2 𝑘𝑘1 +𝑐𝑐𝑐𝑐2 𝑘𝑘2 ) + 𝜖𝜖𝜖𝜖𝜖𝜖𝜖𝜖1 𝑘𝑘2 (𝑝𝑝𝑝𝑝𝑝𝑝, and 𝐶𝐶 𝐶𝐶𝐶21 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝21 𝑘𝑘2 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝21 𝑞𝑞𝑞𝑞2 (𝑝𝑝𝑝𝑝𝑝𝑝. e interior equilibrium point 𝐸𝐸∗ exists if 𝑝𝑝 𝑝𝑝𝑝, 𝑘𝑘1 >𝑥𝑥 and 𝐵𝐵2 ≥ 4𝐴𝐴𝐴𝐴.
𝜆𝜆3 + Ω1 𝜆𝜆2 + Ω2 𝜆𝜆 𝜆𝜆3 = 0,
(15)
Ω1 = − 𝐴𝐴11 + 𝐴𝐴22 ,
Ω2 = 𝐴𝐴11 𝐴𝐴22 − 𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴23 𝐴𝐴32 − 𝐴𝐴13 𝐴𝐴31 ,
Ω3 = − 𝐴𝐴13 𝐴𝐴21 𝐴𝐴32 − 𝐴𝐴22 𝐴𝐴31 − 𝐴𝐴23 𝐴𝐴11 𝐴𝐴32 − 𝐴𝐴12 𝐴𝐴31 . (16) By the Routh-Hurwitz criteria [12], the positive equilibrium point 𝐸𝐸∗ (𝑥𝑥∗ ,𝑦𝑦∗ ,𝑧𝑧∗ ) is locally asymptotically stable if and only if Ω1 > 0, Ω3 > 0, and Ω1 Ω2 − Ω3 > 0 hold. A sufficient condition for local stability of 𝐸𝐸∗ (𝑥𝑥∗ ,𝑦𝑦∗ ,𝑧𝑧∗ ) is given by the following theorem.
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0.0795 0.079 0.0785 0
500
1000
1500
3.3434 Alternative food
3.3432 3.343 3.3428 3.3426
2500
3000
3500
2.2567 2.2566 2.2565 0
500 1000 1500 2000 2500 3000 3500 4000 4500 Time
0
500 1000 1500 2000 2500 3000 3500 4000 4500 Time
3.3424 3.3422
3.344 Predator
Predator population
2000
Time
3.342 3.3418 0.0788
2.2568 2.2569 0.0792 0.0794 2.2565 2.2566 2.2567 Prey popu e food lation Alternativ
0.079
3.342 3.34
F 2: e trajectory and time series diagrams of prey, alternative prey, and predator population of the system for 𝑘𝑘1 = 3.0, 𝑘𝑘2 = 2.5, 𝑝𝑝 𝑝𝑝𝑝𝑝, 𝑞𝑞 𝑞𝑞𝑞𝑞𝑞, 𝑎𝑎 𝑎𝑎𝑎𝑎, 𝑐𝑐 𝑐𝑐𝑐𝑐, ℎ = 0.5, 𝜖𝜖 𝜖𝜖𝜖𝜖, and 𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑.
eorem 2. e interior equilibrium point 𝐸𝐸∗ (𝑥𝑥∗ , 𝑦𝑦∗ , 𝑧𝑧∗ ) for the system (2) is locally asymptotically stable if the following conditions hold: 2
𝑎𝑎𝑎𝑎𝑎𝑎𝑎∗ 𝑧𝑧∗ +𝜖𝜖 𝑞𝑞𝑞𝑞∗ +𝑎𝑎𝑎𝑎 𝑎 𝑎𝑎𝑎 𝑥𝑥∗ 𝑧𝑧∗ 1 + 𝑎𝑎𝑎𝑎∗ + 𝑐𝑐𝑐𝑐𝑐∗ < 0, 2
ℎ𝑞𝑞𝑞𝑞𝑞𝑞∗ 𝑧𝑧∗ +𝜖𝜖 𝑝𝑝𝑝𝑝∗ +𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑐𝑐𝑐𝑐∗ 𝑧𝑧∗ 1 + 𝑎𝑎𝑎𝑎∗ + 𝑐𝑐𝑐𝑐𝑐∗ < 0, ∗
∗
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑎𝑎𝑎𝑎 𝑎 𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 − 𝑐𝑐𝑐𝑐𝑐 > 0.
(17)
Proof. From the above derivation, it is clear that 𝐴𝐴11 < 0, 𝐴𝐴12 > 0, 𝐴𝐴13 < 0, 𝐴𝐴21 > 0, 𝐴𝐴22 < 0, 𝐴𝐴23 < 0, 𝐴𝐴31 > 0, 𝐴𝐴32 > 0, and 𝐴𝐴33 = 0. Under these conditions, it is easy to show that Ω1 > 0 and Ω3 > 0. Now, by calculating Ω1 Ω2 −Ω3 , we get Ω1 Ω2 − Ω3 = (𝐴𝐴11 + 𝐴𝐴22 )(𝐴𝐴23 𝐴𝐴32 + 𝐴𝐴13 𝐴𝐴31 − 𝐴𝐴11 𝐴𝐴22 ) + 𝐴𝐴11 (𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴23 𝐴𝐴32 ) + 𝐴𝐴22 (𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴13 𝐴𝐴31 ) + 𝐴𝐴13 𝐴𝐴21 𝐴𝐴32 + 𝐴𝐴23 𝐴𝐴12 𝐴𝐴31 . Under the conditions (17), it is proven that 𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴23 𝐴𝐴32 < 0, 𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴13 𝐴𝐴31 < 0, and 𝐴𝐴13 𝐴𝐴21 𝐴𝐴32 + 𝐴𝐴23 𝐴𝐴12 𝐴𝐴31 > 0. erefore, we get Ω1 Ω2 − Ω3 > 0. Hence, 𝐸𝐸∗ (𝑥𝑥∗ , 𝑦𝑦∗ , 𝑧𝑧∗ ) is locally asymptotically stable.
Now, I investigate the global stability of the equilibrium point 𝐸𝐸∗ of the system (2). 3.2. Global Stability of Interior Equilibrium Points
eorem 3. Suppose that the positive equilibrium point 𝐸𝐸∗ (𝑥𝑥∗ , 𝑦𝑦∗ , 𝑧𝑧∗ ) is locally asymptotically stable. en it is a globally asymptotically stable if the following condition holds:
2
∗
2
1 + 𝑦𝑦 1 + 𝑥𝑥∗ 𝑑𝑑𝑑𝑑∗ + + 2 2 𝜖𝜖
Research Article Global Stability of Predator-Prey System with Alternative Prey Banshidhar Sahoo Department of Mathematics, Daharpur A.P.K.B Vidyabhaban, Paschim Medinipur, West Bengal, India Correspondence should be addressed to Banshidhar Sahoo; [email protected] Received 1 April 2012; Accepted 7 June 2012 Academic Editors: M. Tamaoki and E. Villegas Copyright © 2013 Banshidhar Sahoo. is 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. A predator-prey model in presence of alternative prey is proposed. Existence and local stability conditions for interior equilibrium points are derived. Global stability conditions for interior equilibrium points are also found. Bifurcation analysis is done with respect to predator’s searching rate and handling time. Bifurcation analysis con�rms the existence of global stability in presence of alternative prey.
1. Introduction e classical predator-prey model based on the logistic growth principle and Hollings predation theory is as follows: 𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 𝑥𝑥 = 𝑥𝑥 1 − − , 𝑑𝑑𝑑𝑑 𝑘𝑘 1 + 𝑐𝑐𝑐𝑐𝑐
(1) 𝑏𝑏𝑏𝑏𝑏𝑏 𝑑𝑑𝑑𝑑 − 𝑑𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 1 + 𝑐𝑐𝑐𝑐𝑐 where 𝑥𝑥 and 𝑦𝑦 represent the density of prey and predator species with carrying capacity 𝑘𝑘. e constant 𝑎𝑎 denotes the food intake rate of predator, 𝑏𝑏 denotes the food conversion rate to predator, and 𝑑𝑑 is the predator’s death rate. e constant 𝑐𝑐, ℎ being predator’s searching rate, and handling time on 𝑥𝑥, respectively. In this model, there is no protection for prey from predator and predator’s survival depends on prey alone. Here the predator species 𝑦𝑦 totally depends on the prey species 𝑥𝑥 and so there is high predation pressure on the prey species. As a result, the prey species has high extinction risk for different searching rate and handling time which is shown in Figure 1. In nature, when the prey population falls below a certain level, the predator searches alternative prey and returns only when the prey population rises to required level. ere are large numbers of three or more species food chain system [1, 2] instead of two species system for the survival of prey species. Van Baalen et al. [3] showed the switching fashion from prey species to alternative prey for persistence of predator-prey system. Plants bene�t from providing food to predators even when
it is also edible to herbivores which is discussed by Van Rijn et al. [4]. Harwood and Obrycki [5] investigated the role of alternative prey in sustaining predator populations. e role of alternative food for biological pest control in predator-prey system is investigated by many scientists [6, 7]. Sahoo [8] studied a food chain model with different functional responses and different growth rates in presence of additional food for construction of real food chain model. Recently, Sahoo [9] showed that additional food is very important for survival of consumer species in an ecosystem. e consequences of providing a predator with additional food and the corresponding effects on the predator-prey dynamics with monotonic and nonmonotonic functional response and its utility in biological control is comparatively studied by Sahoo [10]. But, all of them assumed that the additional food is not dynamic but maintained at a speci�c constant level either by the nature or by an external agency. In this context, I have proposed a predator-prey model with alternative prey (a dynamic additional food for predator). is model is similar to two prey one predator model. e following assumptions are done to formulate the model. (a) Let 𝑥𝑥 be the prey density, let 𝑦𝑦 be the density of alternative prey and the density of the predator is 𝑧𝑧.
(b) Both preys are distributed uniformly in the habitat.
(c) e prey and alternative prey grow as per logistic equation in the absence of predators.
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3.5
2.5
3
Predator
Population density
Population density
c = 0.8
4
c = 0.5
2 1.5
Predator 2.5 2 1.5
1 1 0.5
0.5
Prey
Prey
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Handling time h
Handling time h
F 1: Bifurcation diagram of the prey and predator species with respect to handling time ℎ for 𝑘𝑘 𝑘 𝑘, 𝑎𝑎 𝑎 𝑎𝑎𝑎, 𝑏𝑏 𝑏𝑏𝑏𝑏, and 𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑.
(d) e predator-prey and predator-alternative prey capture rates are of Holling type II. (e) e constant 𝑎𝑎 is predator’s handling time on 𝑥𝑥 and 𝑐𝑐, ℎ are predator’s searching rate and handling time on 𝑦𝑦, respectively.
With the above assumptions, we formulate the following model as 𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑 𝑥𝑥 = 𝑥𝑥 1 − − , 𝑑𝑑𝑑𝑑 𝑘𝑘1 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
𝑦𝑦 𝑞𝑞𝑞𝑞𝑞𝑞 𝑑𝑑𝑑𝑑 = 𝑦𝑦 1 − − , 𝑑𝑑𝑑𝑑 𝑘𝑘2 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 𝜖𝜖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧 = − 𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
dissipativeness of the system. e local stability and global stability of the interior equilibrium points of the system are examined in Section 3. Moreover, we discuss the numerical experiment of our system in Section 4. Finally, conclusion is written in Section 5.
2. Theoretical Analysis 2.1. Positive Invariance. Let 𝑋𝑋 𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑇𝑇 ∈ 𝑅𝑅3 and
(2)
where 𝑘𝑘1 and 𝑘𝑘2 are the carrying capacity of prey (𝑥𝑥) and alternative prey (𝑦𝑦), respectively; the constants 𝑝𝑝 and 𝑞𝑞 are predator’s (𝑧𝑧) food intake rate on prey and alternative prey respectively. e constants 𝜖𝜖 and 𝑐𝑐𝑐𝑐 are conversion rates of prey and alternative food to predator, respectively; 𝑑𝑑 is constant death rate for predator. Here we assume that predator’s food intake rate on prey (𝑥𝑥) is much more greater than that of alternative prey (i.e, 𝑝𝑝 𝑝 𝑝𝑝). e parameters 𝑐𝑐 and ℎ characterize the alternative prey. is formulation implies that the density of prey (𝑥𝑥) and alternative prey (𝑦𝑦) are scaled with respect to search rate of the predators, this can be done without loss of generality. e system has to be analyzed with the following initial conditions: 𝑥𝑥𝑥𝑥𝑥 𝑥 𝑥, 𝑦𝑦𝑦𝑦𝑦𝑦𝑦, 𝑧𝑧𝑧𝑧𝑧𝑧𝑧. e main objective of this paper is to investigate the dynamic properties and behaviors of the system. Here I shall analyze the dynamics of the system with respect to predator’s (𝑧𝑧) searching rate 𝑐𝑐 and handling time (ℎ) on alternative prey 𝑦𝑦. is paper is organized as follows. In Section 2, we show the
𝑥𝑥 1 −
𝑝𝑝𝑝𝑝𝑝𝑝 𝑥𝑥 − 𝑘𝑘1 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
𝐹𝐹1 (𝑋𝑋) 𝑦𝑦 𝑞𝑞𝑞𝑞𝑞𝑞 𝑦𝑦 1 − − 𝐹𝐹 (𝑋𝑋) = 𝐹𝐹2 (𝑋𝑋) = , 𝑘𝑘2 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐹𝐹3 (𝑋𝑋) 𝜖𝜖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧 − 𝑑𝑑𝑑𝑑 1 + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (3)
3 where 𝐹𝐹𝐹𝐹𝐹𝐹 𝐹 𝐹𝐹+ → 𝑅𝑅3 and 𝐹𝐹 𝐹 𝐹𝐹∞ + (𝑅𝑅 ). en system (2) becomes
𝑋𝑋̇ 𝑋𝑋𝑋 (𝑋𝑋)
(4)
with 𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋0 ∈ 𝑅𝑅3+ . It is easy to verify that whenever choosing 𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋3 such that 𝑋𝑋𝑖𝑖 =0 then [𝐹𝐹𝑖𝑖 (𝑋𝑋𝑋𝑋𝑋𝑋𝑖𝑖 =0 ≥0 (for 𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖). Now any solution of (4) with 𝑋𝑋0 ∈ 𝑅𝑅3+ , say 𝑋𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋0 ), is such that 𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋3+ for all 𝑡𝑡 𝑡 𝑡 (Nagumo, [11]). eorem 1. All the solutions of the system (2) which initiate in 𝑅𝑅3+ are uniformly bounded.
Proof. Let (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 be any solution of the system (2) with positive initial conditions.
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Let us consider that 𝑤𝑤 𝑤 𝑤𝑤 𝑤 𝑤𝑤 𝑤 𝑤𝑤𝑤
that is,
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = + + . 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
erefore,
𝑦𝑦 𝑝𝑝𝑝𝑝𝑝𝑝 𝑥𝑥 𝑑𝑑𝑑𝑑 =𝑥𝑥 1 − − +𝑦𝑦 1 − 𝑑𝑑𝑑𝑑 𝑘𝑘1 1 + 𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎 𝑘𝑘2 erefore,
that is,
𝜖𝜖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧 𝑞𝑞𝑞𝑞𝑞𝑞 − + − 𝑑𝑑𝑑𝑑𝑑 1 + 𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎 1 + 𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 ≤ 𝑥𝑥 (1 − 𝑥𝑥) +𝑦𝑦 1 − 𝑦𝑦 − 𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ≤ 2 − 𝜃𝜃 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 , 𝑑𝑑𝑑𝑑
where 𝜃𝜃 𝜃 𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃:
𝑑𝑑𝑑𝑑 + 𝜃𝜃𝜃𝜃𝜃𝜃𝜃 𝑑𝑑𝑑𝑑
(5)
e Jacobian matrix of the system (2) at the interior equilibrium point 𝐸𝐸∗ is
(6) where
𝐴𝐴11 = 1 −
(7)
𝐴𝐴12 =
𝐴𝐴13 =
(8)
𝐴𝐴21 =
𝐴𝐴23 =
(10)
𝐴𝐴31 =
0 < 𝑤𝑤 𝑤
+ 𝑤𝑤 𝑥𝑥 (0) ,𝑦𝑦 (0) ,𝑧𝑧 (0) 𝑒𝑒−𝜃𝜃𝜃𝜃 .
𝜃𝜃
𝐴𝐴32 =
(11)
(13)
𝑝𝑝 1 +𝑐𝑐𝑐𝑐𝑐∗ 𝑧𝑧 2𝑥𝑥∗ − , 2 𝑘𝑘1 1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝∗ 𝑧𝑧∗
2
,
𝑞𝑞𝑞𝑞𝑞𝑞∗ 𝑧𝑧∗
2
,
1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗ −𝑝𝑝𝑝𝑝∗ , 1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
𝐴𝐴22 = 1 −
(9)
Applying the theory of differential inequality we obtain 2 1 − 𝑒𝑒−𝜃𝜃𝜃𝜃
𝐴𝐴11 𝐴𝐴12 𝐴𝐴13 𝐽𝐽 𝐸𝐸∗ = 𝐴𝐴21 𝐴𝐴22 𝐴𝐴23 , 𝐴𝐴31 𝐴𝐴32 𝐴𝐴33
𝑞𝑞 1 + 𝑎𝑎𝑎𝑎∗ 𝑧𝑧 2𝑦𝑦∗ − , 2 𝑘𝑘2 1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
−𝑞𝑞𝑞𝑞∗ , 1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
𝜖𝜖 𝑝𝑝𝑝𝑝∗ +𝑐𝑐 ℎ𝑝𝑝 𝑝𝑝𝑝𝑝𝑝 𝑦𝑦∗ 𝑧𝑧∗ 2
1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
𝜖𝜖𝜖𝜖 𝑞𝑞𝑞𝑞∗ + 𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞 𝑥𝑥∗ 𝑧𝑧∗
𝐴𝐴33 = 0.
2
1 + 𝑎𝑎𝑎𝑎∗ +𝑐𝑐𝑐𝑐𝑐∗
,
(14)
,
For 𝑡𝑡 𝑡 𝑡, we have 0 < 𝑤𝑤 𝑤𝑤𝑤𝑤𝑤. Hence all the solutions of the system (2) that initiate in 𝑅𝑅3+ are con�ned in the region 𝑆𝑆 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑆 𝑆𝑆3+ ∶ 𝑤𝑤 𝑤 𝑤𝑤𝑤𝑤 𝑤 𝑤𝑤, for any 𝜂𝜂 𝜂 𝜂𝜂, which means that all species are uniformly bounded for any initial value in 𝑅𝑅3+ . is proves the theorem.
e characteristic equation of the Jacobian matrix 𝐸𝐸∗ is given by
3. Stability Analysis
where
3.1. Existence and Local Stability of Interior Equilibrium Points. e interior equilibrium point of the system is given by 𝐸𝐸∗ (𝑥𝑥∗ ,𝑦𝑦∗ ,𝑧𝑧∗ ), where, 𝑦𝑦∗ = (𝑘𝑘1 𝑘𝑘2 (𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝2 𝑞𝑞𝑞𝑞∗ )/𝑘𝑘1 𝑝𝑝, 𝑧𝑧∗ = (𝜖𝜖𝜖𝜖𝜖1 −𝑥𝑥𝑥𝑥𝑥𝑥1 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝2 𝑘𝑘1 +𝑐𝑐𝑐𝑐2 𝑘𝑘2 )𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥1 𝑘𝑘2 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝1 𝑝𝑝𝑝, and 𝑥𝑥∗ is the positive root of the equation 2
𝐴𝐴𝐴𝐴∗ + 𝐵𝐵𝐵𝐵∗ + 𝐶𝐶 𝐶𝐶𝐶
(12)
where, 𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴2 𝑘𝑘1 +𝑐𝑐𝑐𝑐2 𝑘𝑘2 ) − 𝑎𝑎𝑎𝑎1 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝2 𝑞𝑞𝑞𝑞, 𝐵𝐵 𝐵𝐵𝐵𝐵𝐵𝐵21 𝑝𝑝 𝑝 𝑘𝑘1 𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝1 𝑘𝑘2 (𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝1 𝑘𝑘2 𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞1 (𝑝𝑝2 𝑘𝑘1 +𝑐𝑐𝑐𝑐2 𝑘𝑘2 ) + 𝜖𝜖𝜖𝜖𝜖𝜖𝜖𝜖1 𝑘𝑘2 (𝑝𝑝𝑝𝑝𝑝𝑝, and 𝐶𝐶 𝐶𝐶𝐶21 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝21 𝑘𝑘2 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝21 𝑞𝑞𝑞𝑞2 (𝑝𝑝𝑝𝑝𝑝𝑝. e interior equilibrium point 𝐸𝐸∗ exists if 𝑝𝑝 𝑝𝑝𝑝, 𝑘𝑘1 >𝑥𝑥 and 𝐵𝐵2 ≥ 4𝐴𝐴𝐴𝐴.
𝜆𝜆3 + Ω1 𝜆𝜆2 + Ω2 𝜆𝜆 𝜆𝜆3 = 0,
(15)
Ω1 = − 𝐴𝐴11 + 𝐴𝐴22 ,
Ω2 = 𝐴𝐴11 𝐴𝐴22 − 𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴23 𝐴𝐴32 − 𝐴𝐴13 𝐴𝐴31 ,
Ω3 = − 𝐴𝐴13 𝐴𝐴21 𝐴𝐴32 − 𝐴𝐴22 𝐴𝐴31 − 𝐴𝐴23 𝐴𝐴11 𝐴𝐴32 − 𝐴𝐴12 𝐴𝐴31 . (16) By the Routh-Hurwitz criteria [12], the positive equilibrium point 𝐸𝐸∗ (𝑥𝑥∗ ,𝑦𝑦∗ ,𝑧𝑧∗ ) is locally asymptotically stable if and only if Ω1 > 0, Ω3 > 0, and Ω1 Ω2 − Ω3 > 0 hold. A sufficient condition for local stability of 𝐸𝐸∗ (𝑥𝑥∗ ,𝑦𝑦∗ ,𝑧𝑧∗ ) is given by the following theorem.
4
ISRN Biotechnology
Prey
0.0795 0.079 0.0785 0
500
1000
1500
3.3434 Alternative food
3.3432 3.343 3.3428 3.3426
2500
3000
3500
2.2567 2.2566 2.2565 0
500 1000 1500 2000 2500 3000 3500 4000 4500 Time
0
500 1000 1500 2000 2500 3000 3500 4000 4500 Time
3.3424 3.3422
3.344 Predator
Predator population
2000
Time
3.342 3.3418 0.0788
2.2568 2.2569 0.0792 0.0794 2.2565 2.2566 2.2567 Prey popu e food lation Alternativ
0.079
3.342 3.34
F 2: e trajectory and time series diagrams of prey, alternative prey, and predator population of the system for 𝑘𝑘1 = 3.0, 𝑘𝑘2 = 2.5, 𝑝𝑝 𝑝𝑝𝑝𝑝, 𝑞𝑞 𝑞𝑞𝑞𝑞𝑞, 𝑎𝑎 𝑎𝑎𝑎𝑎, 𝑐𝑐 𝑐𝑐𝑐𝑐, ℎ = 0.5, 𝜖𝜖 𝜖𝜖𝜖𝜖, and 𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑.
eorem 2. e interior equilibrium point 𝐸𝐸∗ (𝑥𝑥∗ , 𝑦𝑦∗ , 𝑧𝑧∗ ) for the system (2) is locally asymptotically stable if the following conditions hold: 2
𝑎𝑎𝑎𝑎𝑎𝑎𝑎∗ 𝑧𝑧∗ +𝜖𝜖 𝑞𝑞𝑞𝑞∗ +𝑎𝑎𝑎𝑎 𝑎 𝑎𝑎𝑎 𝑥𝑥∗ 𝑧𝑧∗ 1 + 𝑎𝑎𝑎𝑎∗ + 𝑐𝑐𝑐𝑐𝑐∗ < 0, 2
ℎ𝑞𝑞𝑞𝑞𝑞𝑞∗ 𝑧𝑧∗ +𝜖𝜖 𝑝𝑝𝑝𝑝∗ +𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑐𝑐𝑐𝑐∗ 𝑧𝑧∗ 1 + 𝑎𝑎𝑎𝑎∗ + 𝑐𝑐𝑐𝑐𝑐∗ < 0, ∗
∗
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑎𝑎𝑎𝑎 𝑎 𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 − 𝑐𝑐𝑐𝑐𝑐 > 0.
(17)
Proof. From the above derivation, it is clear that 𝐴𝐴11 < 0, 𝐴𝐴12 > 0, 𝐴𝐴13 < 0, 𝐴𝐴21 > 0, 𝐴𝐴22 < 0, 𝐴𝐴23 < 0, 𝐴𝐴31 > 0, 𝐴𝐴32 > 0, and 𝐴𝐴33 = 0. Under these conditions, it is easy to show that Ω1 > 0 and Ω3 > 0. Now, by calculating Ω1 Ω2 −Ω3 , we get Ω1 Ω2 − Ω3 = (𝐴𝐴11 + 𝐴𝐴22 )(𝐴𝐴23 𝐴𝐴32 + 𝐴𝐴13 𝐴𝐴31 − 𝐴𝐴11 𝐴𝐴22 ) + 𝐴𝐴11 (𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴23 𝐴𝐴32 ) + 𝐴𝐴22 (𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴13 𝐴𝐴31 ) + 𝐴𝐴13 𝐴𝐴21 𝐴𝐴32 + 𝐴𝐴23 𝐴𝐴12 𝐴𝐴31 . Under the conditions (17), it is proven that 𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴23 𝐴𝐴32 < 0, 𝐴𝐴12 𝐴𝐴21 − 𝐴𝐴13 𝐴𝐴31 < 0, and 𝐴𝐴13 𝐴𝐴21 𝐴𝐴32 + 𝐴𝐴23 𝐴𝐴12 𝐴𝐴31 > 0. erefore, we get Ω1 Ω2 − Ω3 > 0. Hence, 𝐸𝐸∗ (𝑥𝑥∗ , 𝑦𝑦∗ , 𝑧𝑧∗ ) is locally asymptotically stable.
Now, I investigate the global stability of the equilibrium point 𝐸𝐸∗ of the system (2). 3.2. Global Stability of Interior Equilibrium Points
eorem 3. Suppose that the positive equilibrium point 𝐸𝐸∗ (𝑥𝑥∗ , 𝑦𝑦∗ , 𝑧𝑧∗ ) is locally asymptotically stable. en it is a globally asymptotically stable if the following condition holds:
2
∗
2
1 + 𝑦𝑦 1 + 𝑥𝑥∗ 𝑑𝑑𝑑𝑑∗ + + 2 2 𝜖𝜖
