Abstract
A nonautonomous plant-hare model with impulse is considered. By using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. Though Gao et al. (2014) considered the periodic solutions of plant-hare model, such model with impulses and delay has not been studied in previous paper.
1. Introduction
In recent years, applications of theory differential equations in mathematical ecology have developed rapidly. Various mathematical models have been proposed in the study of population dynamics (see [1–13]). Recently, Gao et al. [5] considered a nonautonomous plant-hare dynamical system with a toxin-determined functional response given by where denotes the density of plant at time , denotes the herbivore biomass at time , is the plant intrinsic growth rate at time , is the per capita rate of herbivore death unrelated to plant toxicity at time , and is the conversion rate at time . is the encounter rate per unit plant, is the fraction of food items encountered that the herbivore ingests, is the carrying capacity of plant, measures the toxicity level, and is the time for handing one unit of plant. , , and are continuously positive periodic functions with period and , , , , and are five positive real constants.
However, birth of many species is an annual birth pulse; for having more accurate description to the system, we need to consider incorporating the impulsive effect into the differential equations. For more biological view of impulses, one can refer to [14–17]. To describe to a system more accurately, we should consider the following impulsive system with delays: where the assumptions on , , , , , , , and are the same as before; , is a strictly increasing sequence with and . We further assume that there exists a such that and for .
Without loss of generality, we will assume for , and ; hence . The main purpose of this paper is to derive easily verifiable sufficient conditions for the existence of multiple positive periodic solutions of (2).
2. Preliminaries
In this section, we cite some definitions and lemmas.
Let denote the space of -periodic functions which are continuous for , are continuous from the left for , and have possible discontinuities of the first kind at points ; that is, the limit from the right of exists but may be different from the value at . We also denote .
For the convenience, we list the following definitions and lemmas.
Definition 1 (see [17]). The set is said to be quasiequicontinuous in if for any there exists a such that if , , , and , then
Lemma 2 (see [17]). The set is relatively compact if and only if (1)is bounded, that is, , for each , and some ;(2) is quasiequicontinuous in .
Lemma 3 (see [17]). Assume that ; then the following inequality holds:
3. Existence of Multiple Positive Periodic Solutions
In this section, sufficient conditions are obtained for the existence of periodic solutions of (2).
In order to obtain the existence of positive periodic solutions of (2), for convenience, we will summarize in the following a few concepts and results from [8] that will be basic for this section.
Let be normed vector spaces, let be a linear mapping, and let be a continuous mapping. The mapping is called a Fredholm mapping of index zero if dimKer and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that and, it follows that is invertible. We 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 4 (see [8]). Let be an open and bounded set. Let be a Fredholm mapping of index zero and let be -compact on . Assume,(a)for each and , ;(b)for each , ;(c).Then has at least one solution in .
Before starting our main result, for the lack of convenience, we denote
The following assumptions are valid throughout this paper: , , . For further convenience, we introduce six positive numbers as follows: where
Under assumptions , , and , it is not difficult to show that
Theorem 5. In addition to (), (), and , suppose that .Then system (2) has at least two positive -periodic solutions.
Proof. Making the change of variables,
Then, system (2) can be rewritten as
Take
and define
Both and are Banach spaces.
Define
,
, ; ,
It is not difficult to show that
;
Since is closed in and and are continuous projectors such that
it follows that is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) exists, which is given by
Then and are defined by
Clearly, and are continuous. By using the Arzela-Ascoli theorem, it is not difficult to prove that is compact for any open bounded set . Moreover, is bounded. Therefore, is -compact on with any open bounded set .
Now, we reach the position to search for an appropriate open, bounded subset for the application of the continuation theorem.
Corresponding to the operator equation, , , we have
Suppose that is a solution of (21) for a certain . Integrating the first equation of (21) over the interval , we obtain
Similarly, integrating the second equation of (21) over the interval , we obtain
It follows from the first equation of (21) and (22) and that
that is,
Similarly, it follows from the second equation of (21) and (23) and that
Since , there exists , such that
From and (23), we see that
which implies
So
This, combined with Lemma 3, gives
or equivalently
In view of and we have
Similarly, it follows from and (23) that
which implies
So
This, combined with Lemma 3, gives
or equivalently
It follows from and that
It follows from (25), (33), and Lemma 3 that
On the other hand, it follows from and (22) and (40) that
It follows from (41) that
This, combined with Lemma 3, gives
Moreover, because of , it follows from (42) that
This, combined with Lemma 3 again, gives
It follows from (44) and (46) that
Obviously, , , , and are independent of .
Now, let us consider with . Note that
Because of (), (), (), and (), we can show that the equation has two distinct solutions:
Choose such that
Let
Then both and are bounded open subsets of . It follows from (2) and (50) that and . With the help of (2), (33), (39), (40), and (47)–(50), it is easy to see that and satisfies the requirement (a) in Lemma 4 for . Moreover, for . A direct computation gives . Here, is taken as the identity mapping since . So far we have proved that satisfies all the assumptions in Lemma 4. Hence, Equation (2) has at least two -periodic solutions. This completes the proof of Theorem 5.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.