Existence of Periodic Solutions for a Delayed Ratio-Dependent Three-Species Predator-Prey Diffusion System on Time Scales
© Zhenjie Liu. 2009
Received: 3 September 2008
Accepted: 21 January 2009
Published: 3 February 2009
This paper investigates the existence of periodic solutions of a ratio-dependent predator-prey diffusion system with Michaelis-Menten functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain suffcient criteria for the existence of periodic solutions for the system. Moreover, when the time scale is chosen as or , the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.
which incorporates mutual interference by predators, where is a Michaelis-Menten type functional response function. Equation (1.1) has been studied by many authors and seen great progress (e.g., see [6–11]).
In order to consider periodic variations of the environment and the density regulation of the predators though taking into account delay effect and diffusion between patches, more realistic and interesting models of population interactions should take into account comprehensively other than one or two aspects. On the other hand, in order to unify the study of differential and difference equations, people have done a lot of research about dynamic equations on time scales. The principle aim of this paper is to systematically unify the existence of periodic solutions for a delayed ratio-dependent predator-prey system with functional response and diffusion modeled by ordinary differential equations and their discrete analogues in form of difference equations and to extend these results to more general time scales. The approach is based on Gaines and Mawhin's continuation theorem of coincidence degree theory, which has been widely applied to deal with the existence of periodic solutions of differential equations and difference equations.
are all rd-continuous positive periodic functions with period ; are nonnegative constants.
which is the discrete time ratio-dependent predator-prey diffusive system of three species with time delays and is also a discrete analogue of (1.6).
A time scale is an arbitrary nonempty closed subset of the real numbers . Throughout the paper, we assume the time scale is unbounded above and below, such as and . The following definitions and lemmas can be found in .
If , then is called right-dense (otherwise: right-scattered), and if , then is called left-dense (otherwise: left-scattered).
If has a left-scattered maximum , then ; otherwise . If has a right-scattered minimum , then ; otherwise .
A function is said to be rd-continuous if it is continuous at right-dense points in and its left-sided limits exists (finite) at left-dense points in . The set of rd-continuous functions will be denoted by .
provided this limit exists, and one says that the improper integral converges in this case.
Definition 2.5 (see ).
One says that a time scale is periodic if there exists such that if , then . For , the smallest positive is called the period of the time scale.
Definition 2.6 (see ).
Let be a periodic time scale with period . One says that the function is periodic with period if there exists a natural number such that , for all and is the smallest number such that .
If , one says that is periodic with period if is the smallest positive number such that for all .
Every rd-continuous function has an antiderivative.
Every continuous function is rd-continuous.
If and , then
if for all , then ;
if on , then .
If , then is nondecreasing.
where is an -periodic function, that is, for all , .
Let be two Banach spaces, let be a linear mapping, and let be a continuous mapping. If is a Fredholm mapping of index zero and there exist continuous projectors and such that , , then the restriction is invertible. Denote the inverse of that map by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exists an isomorphism .
Lemma 2.11 (Continuation theorem ).
for each ;
for each ;
Then the operator equation has at least one solution in .
Lemma 2.12 (see ).
3. Existence of Periodic Solutions
The fundamental theorem in this paper is stated as follows about the existence of an -periodic solution.
then the system (1.4) has at least one -periodic solution.
Take , where is taken sufficiently large such that , and such that each solution of the system satisfies if the system (3.35) has solutions. Now take . Then it is clear that verifies the requirement (a) of Lemma 2.11.
When , is a constant vector in with , from the definition of , we can naturally derive whether the system (3.35) has solutions or not. This shows that the condition (b) of Lemma 2.11 is satisfied.
where is the Brouwer degree. By now we have proved that verifies all requirements of Lemma 2.11. Therefore, (1.4) has at least one -periodic solution in . The proof is complete.
If the conditions in Theorem 3.1 hold, then both the corresponding continuous model (1.6) and the discrete model (1.7) have at least one -periodic solution.
If and in (1.6), then the system (1.6) reduces to the continuous ratio-dependence predator-prey diffusive system proposed in .
where are positive -periodic functions, is nonnegative constant. It is easy to obtain the corresponding conclusions on time scales for the system (3.40).
Suppose that (i) , (ii) hold, then (3.40) has at least one -periodic solution.
The result in Corollary 3.5 is same as those for the corresponding continuous and discrete systems.
The author is very grateful to his supervisor Prof. M. Fan and the anonymous referees for their many valuable comments and suggestions which greatly improved the presentation of this paper. This work is supported by the Foundation for subjects development of Harbin University (no. HXK200716) and by the Foundation for Scientific Research Projects of Education Department of Hei-longjiang Province of China (no. 11513043).
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