Research Article | Open Access
Behavior of the Solutions for Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response on Periodic Time Scales in Shifts
We consider two-dimensional predator-prey system with Beddington-DeAngelis type functional response on periodic time scales in shifts. For this special case we try to find under which conditions the system has -periodic solution.
This study is mainly about the predator-prey dynamic systems with Beddington-DeAngelis type functional response on periodic time scales in shifts. Therefore the main tools that we have used in this study are the time scale calculus, periodic time scales in shifts, predator-prey dynamic systems, and their functional response which shows the effect of predator and prey on each other.
First of all, the main tool we have used is time scales calculus, which first appeared in 1990 in the thesis of Hilger . The main aim of this new topic is to unify the discrete and the continuous dynamic systems; in other words, the unification of dynamical systems obtained from differential equations and the difference equations is the principle target of this new area. After this study, many studies have been done on some properties of dynamical systems on time scale calculus such as  and our study can be seen as the continuation of those studies.
Secondly, the dynamic systems that we have considered in this study are the predator-prey ones which are very important in the mathematical ecology that is the branch of the mathematical biology. Many studies have been done on this type of dynamical systems, since these systems help us to understand the future of the considered species. For instance, in a determined territory, if there are two species (one of them is prey and the other is predator), using such a system which models their life gives us some clues about whether predator or prey goes to extinction or their life cycles are permanent or not.
Predator-prey equations are also known as the Lotka-Volterra equations. The Lotka-Volterra predator-prey model was initially proposed by Lotka in the theory of autocatalytic chemical reactions in 1910 [3, 4]. This was effectively the logistic equation , which was originally derived by Verhulst . In 1920 Lotka extended, via Kolmogorov, the model to “organic systems” using a plant species and a herbivorous animal species as an example  and in 1925 he utilised the equations to analyse predator-prey interactions in his book on biomathematics  arriving at the equations that we know today. Vito Volterra, who made a statistical analysis of fish catches in the Adriatic Sea, independently investigated the equations in 1926.
This model was developed by several researchers in the following years. One of them is Holling who is the first to propose using the idea of functional response in [9, 10]. Both the Lotka Volterra model and Holling’s extensions have been used to model the moose and wolf populations in Isle Royale National Park , which with over 50 published papers is one of the best studied predator-prey relationships. In addition to these, there are many studies that use the predator-prey dynamic systems with Holling type functional responses that study the permanence, stability, periodicity, and such different aspects of these systems. The papers [12–14] can be some of its examples.
After the extention of Holling that is about the effect of predator and prey to each other, Arditi and Gizburg made some changes on the extention of Holling on the functional response and this new functional response is known as the ratio dependent functional response and as derivative of it there is also semi ratio dependent functional responses. Again there are many studies that are about the several structures of the predator-prey dynamic systems such as [12, 15–19].
After that Beddington and DeAngelis proposed another functional response separately, because of the some advantages of that new type of functional response. Nowadays, this is known as Beddington-DeAngelis type functional response. According to the studies [20, 21], the advantages are at low densities; this type of functional response can avoid some of the singular behaviors of ratio dependent models and predator feeding can be described much better over a range of predator-prey abundances by using this functional response. Therefore, we preferred to use this type of functional response in our model. The following studies are some of the examples that investigate the several aspects of this model: [12, 22–25].
In recent years, after the development of time scale calculus, the model of predator-prey dynamic systems models started to adapt to the time scales case because of some aspects of this new calculus. At the very beginning, as it is mentioned, time scale case is the unification of continuous and discrete systems. Because of the unordinary life cycle of some species like insects studying with the time scale model of these dynamical systems becomes important. When we consider the life cycle of an insect, most of them live in the summer and then die and their eggs become dormant in the winter. Thus, the life cycle of an insect contains both continuous and discrete time intervals. For such a system, using the model that is obtained by the time scale is more appropriate. The papers [12, 26, 27] are some examples for the studies that are done on the predator-prey dynamic systems on time scale calculus.
To investigate the periodic solutions on time scale case of the predator-prey dynamic systems, the notion of periodic time scale becomes important which is defined as follows: if the given time scale is -periodic, then for each , also There are several papers such as [12, 24, 27] that study the -periodic solutions of the predator-prey models. However, since there are many different kinds of species in the world this periodicity notion on an arbitrary time scale needs some development. This was first done by Adivar in his study  and we meet with the notion periodic time scales in shifts. According to the suggestion of Bohner in the conference PODE 2014, we started to study on the predator-prey dynamic systems with Beddington-DeAngelis type functional response with periodic time scales in shifts and we obtain the following results.
Theorem 1 (continuation theorem, ). Let be a Fredholm mapping of index zero and let be -compact on . Assume the following:(a)For each , any satisfying is not on ,; that is, .(b)For each and the Brouwer degree Then has at least one solution lying in .
We will also give the following lemma, which is essential for this paper.
Definition 2 (see ). Let the time scale include a fixed number where is a nonempty subset of , such that there exist operators which satisfy the following properties:(P.1)With respect to their second arguments the functions are strictly increasing; that is, if then(P.2)If with , then , and if with , then .(P.3)If , then and Moreover, if , then and holds.(P.4)If , then and , respectively.(P.5)If and , then and , respectively.Then the backward operator is and the forward operator is which are associated with (called the initial point). Shift size is the variable in . The values and in indicate units translation of the term to the right and left, respectively. The sets are the domains of the shift operators , respectively.
Definition 3 (see ). Let be a time scale with the shift operators associated with the initial point . The time scale is said to be periodic in shifts if there exists such that for all Furthermore, if then is called the period of the time scale .
Definition 4 (periodic function in shifts and , ). Let be a time scale that is periodic in shifts and with the period . We say that a real valued function defined on is periodic in shifts if there exists such that The smallest number such that it is called the period of .
Notation 1. Consider
Lemma 5. Let our time scale be periodic in shifts and for each is constant. Then is also constant , where for and = Here is a periodic function in shifts.
Proof. We get the desired result, if we are able to show that for any (): Since is a periodic time scale in shifts (WLOG ) there exits such that Hence it is also enough to show that Because of the definition of the time scale and , and for each By using change of variables we get the result. If , then by the assumption of the lemma When , then and when , then : Hence proof follows.
3. Main Result
The equation that we investigate is
In (9), let , , , , , , , , and , , . , , , and , such that for and Each function is from
Lemma 6. Let and . is defined as in Lemma 5. If is periodic function in shifts, then
Proof. We only show the first inequality as the proof of the second inequality is similar to the proof of the other one. Since is periodic function in shifts, without loss of generality, it suffices to show that the inequality is valid for If then the first inequality is obviously true. If Therefore
If which gives
The proof is complete.
Remark 7 (see ). Consider the following equation:This is the predator-prey dynamic system that is obtained from ordinary differential equations. Let . In (9), by taking and , we obtain equality (13), which is the standard predator-prey system with Beddington-DeAngelis functional response. Many studies have been done on this system and [22, 26, 29] are some of their examples.
Let By using equality (9), we obtain Here again by taking and , we obtainwhich is the discrete time predator-prey system with Beddington De-Angelis type functional response and also the discrete analogue of (13). This system was studied in [23, 30, 31]. Since (9) incorporates (13) and (15) as special cases, we call (9) the predator-prey dynamic system with Beddington-DeAngelis functional response on time scales.
For (9), and denote the density of prey and the predator. Therefore and could be negative. By taking the exponentials of and , we obtain the amount of prey and predators that are living per unit of an area. In other words, for the general time scale case, our equation is based on the natural logarithm of the density of the predator and prey. Hence and could be negative.
For (13) and (15), since and , the given dynamic systems directly depend on the density of the prey and predator.
Theorem 8. Assume that for the given time scale while T is constant, is equal for each In addition to conditions on coefficient functions and Lemma 5 if andare satisfied then there exist at least one -periodic solution.
Proof. with the norm with the normLet us define the mappings and by such that and such that Then , where and are constants: is closed in It is obvious that . To show , we have to prove that It is obvious that when we take an element from and an element from , we find an element of by summing these two elements. If we take an element , and WLOG taking we have , where is a constant. Let us define a new function Since is constant by Lemma 5 if we take the integral of from to , we get Similar steps are used for . can be written as the summation of an element from and an element from . Also it is easy to show that any element in is uniquely expressed as the summation of an element and an element from . We get the desired result, since codim is 2. Hence is a Fredholm mapping of index zero. There exist continuous projectors and such that The generalized inverse is given, Let Clearly, and are continuous. Here and are Banach spaces. Since for the given time scale while is constant, is equal for each then we can apply Arzela-Ascoli theorem and by using Arzela-Ascoli theorem we can find is compact for any open bounded set Additionally, is bounded. Thus, is -compact on with any open bounded set
To apply the continuation theorem we investigate the following operator equation:Let be any solution of system (26). Integrating both sides of system (26) over the interval we obtainFrom (26) and (27) we getSince , then there exist , such thatIf is the minimum point of on the interval because is a function that is periodic in shifts for any on the interval the minimum point of is and We have similar results for the other points for
By the first equation of (27) and (29) Since we get Using the second inequality in Lemma 6 we haveBy the first equation of (27) and (29) Then we get using the first inequality in Lemma 6 we haveBy (32) and (35) From the second equation of (27) and the second equation of (32), we can derive that ThereforeBy the assumption of Theorem 8 we getHence, by using the first inequality in Lemma 6 and the second equation of (27),Again using the second equation of (27) we obtain Using the assumption of Theorem 8 we obtain By using the second inequality in Lemma 6,By (39) and (42) we have . Obviously, and are both independent of Let . Then Let ; then verifies requirement (a) in Theorem 1. When , is a constant with , thenwhere is the identity operator.
Let us define the homotopy such that , where Take as the determinant of the Jacobian of Since , then Jacobian of is All the functions in Jacobian of are positive; then is always positive. Hence Thus all the conditions of Theorem 1 are satisfied. Therefore system (9) has at least a positive -periodic solution.
There are many studies about the predator-prey dynamic systems on time scale calculus such as [12, 17, 27, 32]. All of these cited studies are about the periodic solutions of the considered system on a periodic time scale. However, in the world, there are many different species. While investigating the periodicity notion of the different life cycle of the species, the -periodic time scales could be a little bit restricted. Therefore, if the life cycle of this kind of species is appropriate to the functional response Beddington De-Angelis, then the results that we have found in that study become more useful and important.
In addition to these, the -periodic solutions for predator-prey dynamic systems with Holling type functional response, semi ratio dependent functional response, and monotype functional response can be also taken into account for future studies. In that dynamic system, delay conditions and impulsive conditions can also be added for new investigations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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