Existence of Multiple Positive Periodic Solutions to Two Species Parasitical Model with Impulsive Effects and Harvesting Terms
By applying Mawhin’s continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of four positive solutions for two species parasitical system with impulsive effects and harvesting terms. Finally, an example is given to illustrate the effectiveness of our results.
In recent years, the existence of periodic solutions in biological models has been widely studied. Models with harvesting terms are often considered. Generally, the model with harvesting terms is described as follows: where and are functions of two species, respectively; and are harvesting terms standing for the harvests (see [1, 2]). Because of the effect of changing environment such as the weather, season, and food, the number of species population periodically varies with the time. The rate of change usually is not a constant. Motivated by this, we consider the periodic nonautonomous population models. For example, two species parasitical system with harvesting terms is as follows : where and denote the densities of the host and the parasites, respectively; , and are all positive continuous functions and denote the intrinsic growth rate, death rate, obtaining nutriment rate from the host, and harvesting rate, respectively. In the model (2), the parasitical influence on its host is negligible. As we know, in population dynamics, many evolutionary processes experience short-time rapid chance after undergoing relatively long sooth variation. Examples include stocking of species and annual immigration. Incorporating these phenomena gives us impulsive differential equations. For the theory of impulsive differential equations, we refer the reader to [4, 5].
However, to the best of our knowledge, there are few results on the existence of multiple periodic solutions for the delay parasitical with impulsive effects in the literatures. This motivates us to consider the existence of multiple periodic solutions for following parasitical with impulsive effects and harvesting terms nonautonomous model: where . is a strictly increasing sequence with and . is the th species population density. denotes the intrinsic growth rate; and stand for death rate, obtaining nutriment rate from the host, and harvesting rate, respectively. represents obtaining nutriment rate from the host; stands for the time-lag in the process of transformation from the th species to the th species. represents the time-lag in the process of intraspecific competition.
In addition, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Therefore, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (e.g, seasonal effects of weather, food supplies, mating habits, etc.), which leads us to assume that , and are all continuous -periodic functions. For impulsive effects, we further assume that there exists a such that and .
Since a very basic and important problem in the study of a population growth model with a periodic environment is the global existence and stability of a positive periodic solution, which plays a similar role as a globally stable equilibrium does in an autonomous model, also, on the existence of positive periodic solutions to system (3), few results are found in the literatures. This motivates us to investigate the existence of a positive periodic or multiple positive periodic solutions for system (3). In fact, it is more likely for some biological species to take on multiple periodic change regulations and have multiple local stable periodic phenomena. Therefore, it is essential for us to investigate the existence of multiple positive periodic solutions for population models. Our main aim of this paper is by using Mawhin’s continuation theorem of coincidence degree theory to establish the existence of four positive periodic solutions for system (3). For the work concerning the multiple existence of periodic solutions of periodic population models which was done using coincidence degree theory, we refer the reader to [6–8].
This paper is organized as follows. In Section 2, by using the continuation theorem of coincidence degree theory and the skills of inequalities, we establish the existence of at least four positive periodic solutions of system (3). An example is presented in the last section to illustrate the effectiveness of our results.
2. Existence of at Least Four Positive Periodic Solutions
We first summarize a few concepts from the book by Gaines and Mawhin .
Let and be real normed vector spaces. Let be a linear mapping and a continuous mapping. The mapping will be called a Fredholm mapping of index zero if dim = codim, and is closed in . If is a Fredholm mapping of index zero, then there exist continuous projectors and such that and , and . It follows that is invertible, and its inverse is denoted by . If is a bounded open subset of , the mapping is called -compact on , and if is bounded, is compact. Because is isomorphic to , there exists an isomorphism .
Lemma 1 (see ). Let be a Fredholm mapping of index zero, and let be -compact on . Assume (a)for each , every solution of is such that ; (b) for each ; (c).
Then has at least one solution in .
Let be a given positive constant, and a finite number of points of the sequence lies in the interval . Let be the set of functions which are piecewise continuous in and have points of discontinuous , where they are continuous from the left. In the set introduce the norm with which becomes a Banach space with the uniform convergence topology.
Definition 2. The set is said to be quasiequicontinuous in , if for any , there exists such that if and , then .
The following result called compactness criterion gives a necessary and sufficient condition for relative compactness in .
Lemma 3 (see ). The set is relatively compact if and only if(a) is bounded, namely, , for each and some ;(b) is quasiequicontinuous in .
For the sake of convenience, we denote ; here is a continuous -periodic function. 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 . Obviously, is continuous if .
For simplicity, we need to introduce some notations as follows: where .
Throughout this paper, we need the following assumptions: ; .
The following results will play an important role in the proof of our main result.
Lemma 4 (see ). Suppose , then
Lemma 5. Let , and ; for the functions and , the following assertions hold. (1) and are monotonically increasing and monotonically decreasing on the variable , respectively. (2) and are monotonically decreasing and monotonically increasing on the variable , respectively. (3) and are monotonically decreasing and monotonically increasing on the variable , respectively.
Proof. In fact, for all and , we have By the relationship of the derivative and the monotonicity, the above assertions obviously hold. The proof of Lemma 5 is complete.
Lemma 6. Assume that and hold, then one has the following inequalities: (1); (2); (3).
Applying Lemma 5, we have
Thus, we have . The proof of (1) of Lemma 6 is complete. For (2) of Lemma 6, we similarly have which imply that ; that is, the inequality (2) of Lemma 6 holds. Let us now prove the inequality (3) of Lemma 6. In fact, since we have
If , then , and
If , then , and Hence, we drive . From the above, we have completed the proof of Lemma 6.
Theorem 7. Assume that and hold. Then system (3) has at least four positive -periodic solutions.
Proof. By making the substitution
system (3) can be reformulated as
Let and define Then both and are Banach spaces. Let where Obviously, Since is closed in , and ; we know that is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) is given byThen where are very similar to that of . By the Lebesgue convergence theorem, and are continuous. Moreover, because of periodicity, it follows from Lemma 3 that are relatively compact for any open bounded set . Thus, is -compact on for any open bounded set .
In order to use Lemma 1, we have to find at least four appropriate open bounded subsets of . Considering the operator equation , we have Assume that is a -periodic solution of system (23) for some . Integrating (23) from to , we have Then Furthermore, note that . There exists , such that .
On the one hand, according to the first equation of (23) and (25), we have In the light of the first equation of (25) and (26) and Lemma 4, we get namely, which implies that Applying Lemma (23), we drive According to (23) and (25), we get
By (25) and (31), we have that is, which implies that Applying Lemma 4, we drive
On the other hand, in the light of the first equation of (25) and (26), we have namely, which implies that that is, which implies that By (25) and (31), we have namely, which implies that that is, which implies that
In view of (30)–(45) and Lemma 6, we have or which imply that, for all ,
Clearly, , and are independent of . Now let Obviously, the number of the above sets is four. We denote these sets as , and . are bounded open subsets of . Thus and satisfies the requirement in Lemma 1.
Now we show that of Lemma 1 holds; that is, we prove when , and . If it is not true, then when , and , constant vector with , and satisfies that is, Similar to the process of (25)–(48), we obtain Then belongs to one of and . This contradicts the fact that and . This proves that in Lemma 1 holds.
Finally, in order to show that in Lemma 1 holds, we only prove that for , and , then it holds that . To this end, we define the mapping by here is a parameter, and is defined by We show that for , and , then it holds that . Otherwise, parameter and constant vector satisfy , that is, that is, Following the argument of (25)–(48), we obtain Equation (57) gives that belongs to one of , and . This contradicts the fact that , and . This proves holds. Note that the system of algebraic equations, has four distinct solutions since and hold: where Similar to the proof of Lemma 6, it is easy to verify that
Therefore uniquely belongs to the corresponding . Since , we can take . A direct computation gives, for , and , Since .