Existence and global stability of periodic solutions of a discrete ratio-dependent food chain model with delay☆
Introduction
Recently, there has been considerable interest in ratio-dependent predator–prey model (see [3], [13], [14], [15], [16] and the references therein). In their paper [15], Hsu, Hwang and Kuang consider the following ratio-dependent food chain model.where x, y and z represent the population density of prey, predator and top predator, respectively. Observe that the simple relation of these three species: z prey on y and only on y, and y prey on x and nutrient recycling is not accounted for. They show that this model is rich in boundary dynamics and is capable of generating extinction dynamics. Specifically, they provide partial answers to question such as: under what scenarios a potential biological control may be successful, and when it may fail.
Since the variation of the environment plays an important role in many biological and ecological systems. In particular, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, the assumption of periodicity of the parameters in the way (in a way) incorporates the periodicity of the environment (e.g., seasonal affects of weather, food supplies, mating habits, etc.). In fact, it has been suggested by Nicholson [9] that any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes. In view of this it is realistic to assume that the parameters in the models are periodic functions of period ω. Therefore, In [12], we study the following nonautonomous delayed ratio-dependent food chain modelwhere r(t),b(t),c1(t),c2(t),d1(t),d2(t),m1(t),m2(t)∈C(R,R+),R+=(0,+∞) are ω-periodic function; τi(t), i=1, 2, 3, a1 and a2 are positive constants. By using the continuation theorem base on Gaines and Mawhin's coincidence degree, sufficient and realistic conditions are obtained for the global existence of positive periodic solutions of system (1.2).
On the other hand, many authors [1], [2], [6], [8] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. However, no such work has been done for ratio-dependent food chain model.
In this paper we consider the following discrete ratio-dependent food chain model with delay l1, l2, l3.which can be looked as a discrete analogue of system (1.2), where and are ω periodic, i.e.,for any k∈Z, where Z, R+ denote the sets of all integers, and nonnegative real numbers, respectively. For biological reasons, we only consider the following initial condition.
A very basic and important ecological problem associated with the study of multispecies population interaction in a periodic environment is the global existence of positive solution which plays the role played by the equilibrium of the autonomous models. The main purpose of this paper is to derive easily verifiable sufficient conditions for the global existence of positive periodic solutions of systems (1.3) and these sufficient conditions are similar to those for corresponding continuous system [12]. By means of constructing suitable Lyapunov function, sufficient conditions for the global stability of positive periodic solutions of systems (1.3) are also obtained. The method used here will be the coincidence degree theory developed by Gaines and Mawhin [7] and Lyapunov function. Such approach was adopted in [4], [10], [11], [17], [18].
Section snippets
Existence of a positive periodic solution
In order to obtain the existence of a positive periodic solution of the system (1.3), we first make the following preparations.
Let X and Y be two Banach spaces. Consider an operator equationwhere is a linear operator and λ is a parameter. Let P and Q denote two projectors such that
Denote by is an isomorphism of onto . Recall that a linear mapping with and , will be called a Fredholm mapping
Global attractivity of a positive periodic solution
In this section, we derive sufficient conditions which guarantee that the positive ω-periodic solution of (1.3) is globally stable. Our strategy in the proof of the global stability of the positive ω -periodic solution of (1.3) is to construct suitable Lyapunov functions. Theorem 3.1 In addition to the assumptions made in Theorem 2.1, assume further that There exist positive constant ν and positive constants ci such that
References (19)
- et al.
Effects of spatial grouping on the functional response of predators
Theoret. Population Biol.
(1999) - et al.
Periodicity in a delayed ratio-dependent predator–prey system
J. Math. Anal. Appl.
(2001) - et al.
Periodic solution of a discrete time nonautonomous ratio-dependent predator–prey system
Math. Comput. Modell.
(2002) - et al.
Periodic solutions of periodic delay Lotka–Volterra equations and systems
J. Math. Anal. Appl.
(2001) - R.P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications, Monographs and textbooks in Pure...
- et al.
Advance Topics in Difference Equations
(1997) Deterministic Mathematics Models in Population Ecology
(1980)- et al.
Coincidence Degree and Nonlinear Differential Equations
(1977) Mathematical Biology
(1989)
Cited by (18)
Global existence of bifurcated periodic solutions in a commensalism model with delays
2012, Applied Mathematics and ComputationAn impulsive ratio-dependent n + 1-species predator-prey model with diffusion
2010, Nonlinear Analysis: Real World ApplicationsPermanence and global stability in a discrete n-species competition system with feedback controls
2008, Nonlinear Analysis: Real World ApplicationsCitation Excerpt :There is a rich literature on this topic. For results on models described by differential equations, see, for example, [1–6,14,20–22,24,25,27–29] and the references therein, while for results on models governed by difference equations, to mention a few, see [8,11–13,15–19,23,26,30–32] and the references therein. Moreover, as we know, ecosystems in the real world are often disturbed by unpredictable forces which can result in changes in biological parameters such as survival rates.
Periodic solutions of a discrete two-species competitive model with stage structure
2008, Mathematical and Computer ModellingPermanence of species in nonautonomous discrete Lotka-Volterra competitive system with delays and feedback controls
2008, Journal of Computational and Applied MathematicsCitation Excerpt :In recent years, great attention has been paid to the dynamic behaviors for the single species or multi-species competitive system of differential equations with feedback control, and many excellent results are obtained [4–6,8,13,14,19,22,23]. But, it should be mentioned here that during the last decade, many scholars had also done works on the ecosystem of population models governed by difference equations, see [7,10,12,15,18,21,24,25] and the references cited therein. This paper is organized as follows.
Permanence and global attractivity of the food-chain system with Holling IV type functional response
2007, Applied Mathematics and Computation
- ☆
This work is supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z), the Foundation for University Key Teacher by the Ministry of Education of China, the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China, the Key Research and Development Program for Outstanding Groups of Lanzhou University of Technology and the Development Program for Outstanding Young Teachers in Lanzhou University of Technology.