Recent Developments and Applications on Discrete Fractional Equations and Related TopicsView this Special Issue
Periodic Solutions of a Nonautonomous Plant-Hare Model with Impulses
A plant-hare model subjected by the effect of impulses is studied in this paper. Sufficient conditions are obtained for the existence of at least one positive periodic solution.
Classical predator-prey model has been well studied (e.g., see [1–8] and the references cited therein). To explore the impact of plant toxicity on the dynamics of plant-hare interactions, Gao and Xia  consider a nonautonomous plant-herbivore dynamical system with a toxin-determined functional response: 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 , 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. To explore the impact of environmental factors (e.g., seasonal effects of weather, food supplies, mating habits, harvesting, etc.), the assumption of periodicity of parameters is more realistic and important. To this reason, they assumed that , , and are continuously positive periodic functions with period and , , , , are five positive real constants.
However, birth of many species is an annual birth pulse, for having more accurate description of the system, we need to consider using the impulsive differential equations. To see how impulses affect the differential equations, for examples, one can refer to [10–17]. Motivated by the above-mentioned works, in this paper, we consider the above system with impulses: 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 .
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 ). The set is said to be quasi-equicontinuous in if for any there exists a such that if ; ; and , then
Lemma 2 (see ). The set is relatively compact if and only if (1) is bounded, that is, , for each , and some ;(2) is quasi-equicontinuous in .
Lemma 3 (see ). Assume that , then the following inequality holds: Before starting the main result, for the sake of convenience, one denotes
3. Existence of Positive Periodic Solutions
Let , be normed vector spaces, let be a linear mapping, and a continuous mapping. The mapping is called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero, there exist continuous projectors and such that , . It follows that is invertible. We denote the inverse of that map by . If is an open bounded subset of , then the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exists an isomorphism .
Lemma 4 (see ). Let be an open and bounded set. Let be a Fredholm mapping of index zero and let be -compact on . Assume(a) for each , , ;(b) for each , ;(c). Then has at least one solution in .
If is a continuous -periodic function, then we set The following assumptions are valid throughout this paper:, , .
For convenience, we introduce two numbers as follows: where .
Theorem 5. In addition to (), (), suppose that . Then system (2) has at least one positive -periodic solution.
Remark 6. If the impulsive operators disappear, then . Then Theorem 5 reduces to the main results in Gao and Xia . This implies that our result generalizes the previous one. It shows that the impulses do affect the system indeed.
Proof. Making the change of variables
Then, system (2) can be rewritten as
Both and are Banach spaces.
Define , , ; , It is not difficult to show that Since is closed in , 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 (see ), 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 is a solution of (19) for a certain . Integrating the first equation of (19) over the interval , we obtain Similarly, integrating the second equation of (19) over the interval , we obtain It follows from the first equation of (19) and (20) and that That is, Similarly, it follows from the second equation of (19) and (21) and that Since , there exists such that
From (20), we see that which implies So This, combined with (23), gives Similarly, it follows from (21) that which implies It follows from that This, combined with (23), gives It follows from (29) and (33) that
On the other hand, it follows from (21) and (34) that which implies It follows from () that This, combined with (25), gives
Similarly, it follows from (21) and (34) that which implies So This, combined with (25), gives It follows from (38) and (42) that Now, let us consider with . Note that It follows from , , and that , which implies that the equation has only one solution Choose such that Set ; then . Let It is clear that verifies the requirement (a) in Lemma 4. When , is a constant with . Then for . Simple computation shows that . Here, is taken as the identity mapping since .
This work is supported by NNSFC and the Natural Science Foundation of Fujiang Province (2013J01010).
L. Chen, Mathematical Models and Methods in Ecology, Science Press, Beijing, China, 1998, (in Chinese).
Y.-H. Xia, “Periodic solution of certain nonlinear differential equations: via topological degree theory and matrix spectral theory,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 22, no. 8, Article ID 940287, 2012.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, UK, 1993.View at: MathSciNet
R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.View at: MathSciNet