Existence and global stability of positive periodic solutions of a discrete predator-prey system with delays
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We study the existence and global stability of positive periodic solutions of a periodic discrete predator-prey system with delay and Holling type III functional response. By using the continuation theorem of coincidence degree theory and the method of Lyapunov functional, some sufficient conditions are obtained. 1. Introduction Many realistic problems could be solved on the basis of constructing suitable mathematical models, but it is obvious that a perfect model cannot be achieved because even if we could put all possible factors in a model, the model could never predict ecological catastrophes or mother nature caprice. Therefore, the best we can do is to look for analyzable models that describe as well as possible the reality on populations. From a mathematical point of view, the art of good modelling relies on the following: (i) a sound understanding and appreciation of the biological problem; (ii) a realistic mathematical representation of the important biological phenomena; (iii) finding useful solutions, preferably quantitative; (iv) a biological interpretation of the mathematical results in terms of insights and predictions. Usually a mathematical model could be described by two types of systems: a continuous system or a discrete one. When the size of the population is rarely small or the population has nonoverlapping generations, we may prefer the discrete models. Among all the mathematical models, the predator-prey systems play a fundamental and crucial role (for more details, we refer to [3, 6]). In general, a predator-prey system may have the form
x − ϕ(x)y, K y = y µϕ(x) − D ,
x = rx 1 −
(1.1)
where ϕ(x) is the functional response function. Massive work has been done on this issue. We refer to the monographs [4, 10, 18, 20] for general delayed biological systems and to Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:4 (2004) 321–336 2000 Mathematics Subject Classification: 34C25, 39A10, 92D25 URL: http://dx.doi.org/10.1155/S1687183904401058
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Existence and stability of periodic solutions
[2, 8, 9, 11, 21, 24] for investigation on predator-prey systems. Here, ϕ(x) may be different response functions: standard type II and type III response functions (Holling [12]), Ivlev’s functional response (Ivlev [17]), and Rosenzweig functional response (Rosenzweig [22]). Systems with Holling-type functional response have been investigated by many authors, see, for example, Hsu and Huang [13], Rosenzweig and MacArthur [22, 23]. They studied the stability of the equilibria, existence of Hopf bifurcation, limit cycles, homoclinic loops, and even catastrophe. On the other hand, in view of the periodic variation of the environment (e.g., food supplies, mating habits, seasonal affects of weather, etc.), it would be of interest to study the global existence and global stability of positive solutions for periodic systems [18]. Recently, some excellent existence results have been obtained by using the coincidence degree method (see, e.g., [5, 14, 15, 16, 19, 27]). Motiva
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