- Open Access
Four periodic solutions for a food-limited two-species Gilpin-Ayala type predator-prey system with harvesting terms on time scales
© Fang and Wang; licensee Springer 2013
- Received: 25 January 2013
- Accepted: 23 August 2013
- Published: 23 September 2013
By using Mawhin’s coincidence degree theory, this paper establishes a new criterion on the existence of four periodic solutions for a food-limited two-species Gilpin-Ayala type predator-prey system with harvesting terms on time scales. An example is given to illustrate the effectiveness of the result.
The theory of calculus on time scales was initiated by Hilger  in order to unify continuous and discrete analysis, and it has become an effective approach to the study of mathematical models involving the hybrid discrete-continuous processes. Since the population dynamics in the real world usually involves the hybrid discrete-continuous processes, it may be more realistic to consider population models on time scales .
In recent years, some researchers studied the existence of periodic solutions for some population models on time scales under the assumption of periodicity of the parameters by using Mawhin’s coincidence degree theory (see [3–7]). To our best knowledge, few papers deal with the existence of multiple periodic solutions for population models with harvesting terms on time scales. The main difficulty is that the techniques used in continuous population models with harvesting terms are generally not available to population models with harvesting terms on time scales. Indeed, almost all papers involving continuous population models with harvesting terms used Fermat’s theorem on local extrema of differentiable functions in real analysis; for example, see [8–11]. However, Fermat’s theorem is not true in time scales calculus.
where and denote the prey and the predator, respectively; (), (), () are all positive continuous functions denoting the intrinsic growth rate, the intra-specific competition rates and the harvesting rates, respectively; is the predation rate of the predator and represents the conversion rate; () are the population numbers of two species at saturation, respectively. () represent a nonlinear measure of interspecific interference. When (), () are the rate of replacement of mass in the population at saturation (including the replacement of metabolic loss and of dead organisms). In this case, system (1.2) is a food-limited population model. For other food-limited population models, we refer to [12–17] and the references cited therein. When (), system (1.2) is a Gilpin-Ayala type population model. Gilpin-Ayala type population models were firstly proposed by Gilpin and Ayala in . For some recent work, we refer to [17, 19–22]. When , (), system (1.2) was consider by Zhao and Ye .
Definition 2.1 
A time scale is an arbitrary nonempty closed subset of the real numbers ℝ.
Let . Throughout this paper, the time scale is assumed to be ω-periodic, i.e., implies . In particular, the time scale under consideration is unbounded above and below.
Definition 2.2 
respectively. If , then t is called right-dense (otherwise, right-scattered), and if , then t is called left-dense (otherwise, left-scattered).
Definition 2.3 
In this case, is called the delta (or Hilger) derivative of f at t. Moreover, f is said to be delta or Hilger differentiable on if exists for all .
Definition 2.4 
Definition 2.5 
A function is said to be rd-continuous if it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of rd-continuous functions will be denoted by .
The following notation will be used throughout this paper.
where is a nonnegative ω-periodic real function, i.e., for all .
Lemma 2.1 
Every rd-continuous function has an antiderivative.
Lemma 2.2 
- (i)If f is continuous at t and t is right-scattered, then f is differential at t with
- (ii)If f is right-dense, then f is differential at t iff the limit
Lemma 2.3 
Lemma 2.4 
is uniformly bounded on ;
is uniformly bounded on .
Then there is a subsequence of converging uniformly on .
We first briefly state Mawhin’s coincidence degree theory (see ).
Let X, Z be normed vector spaces, be a linear mapping, be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors (i.e., linear and idempotent linear operators) and such that , . If we define as the restriction of L to , then is invertible. We denote the inverse of that map by . If Ω is an open bounded subset of X, the mapping N will be called L-compact on if is bounded and is compact, i.e., continuous and such that is relatively compact. Since ImQ is isomorphic to KerL, there exists an isomorphism .
For convenience, we introduce Mawhin’s continuation theorem  as follows.
for every and every ;
for every ;
Brouwer degree .
Then has at least one solution in .
Lemma 3.2 
From now on, we always assume that:
(H1) , , , (), () are positive continuous ω-periodic functions, () are positive constants.
- (1)There exist such that
- (2)There exist such that
- (3)There exist such that
By assertions (1)-(3), assertion (4) holds.
which implies that assertion (5) also holds. □
Now, we are ready to state the main result of this paper.
Theorem 3.1 Assume that (H1)-(H3) hold. Then system (1.1) has at least four ω-periodic solutions.
Equipped with the above norm , it is easy to verify that X and Z are both Banach spaces.
We first show that L is a Fredholm mapping of index zero and N is L-compact on for any open bounded set . The argument is standard, one can see [3–5]. But for the sake of completeness, we give the details here.
Obviously, QN and are continuous. By Lemma 2.4, it is not difficult to show that is compact for any open bounded set . Moreover, is bounded. Hence, N is L-compact on for any open bounded set .
In order to apply Lemma 3.1, we need to find at least four appropriate open, bounded subsets , , , in X.
Here, J is taken as the identity mapping since . So far we have proved that satisfies all the assumptions in Lemma 3.1. Hence, (1.1) has at least four ω-periodic solutions () with . Obviously, () are different. □
Hence, all the conditions in Theorem 3.1 are satisfied. By Theorem 3.1, system (1.1) has at least four 2-periodic solutions.
This research is supported by the National Natural Science Foundation of China (Grant No. 10971085).
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