JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 10.1155/2016/7308609 7308609 Research Article Stability Analysis of the Periodic Solutions of Some Kinds of Predator-Prey Dynamical Systems http://orcid.org/0000-0003-1853-3959 Pelen Neslihan Nesliye 1 Jiwari Ram Department of Mathematics Ondokuz Mayis University Samsun Turkey omu.edu.tr 2016 10112016 2016 29 09 2016 18 10 2016 2016 Copyright © 2016 Neslihan Nesliye Pelen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Analysis of predator-prey dynamical systems that have the functional response which generalizes the other types of functional responses in two dimensions is mainly studied in this paper. The main problems for this study are to detect the if and only if conditions for attaining the periodic solution of the considered system and to find the condition for global asymptotic stability of this solution for some different types of predator-prey systems that are obtained from that system. To get the desired results, some aspects of semigroup theory for stability analysis and coincidence degree theory are used.

1. Introduction

Predator-prey dynamic systems mainly investigate the relationships between the species and the relation of the species with the outer environment. Analysis of such kind of mathematical models is really important because, by using these analytical results, one can see the future of the species. The analytical results for these systems change according to two main issues. The first one is the functional response, which shows the effect of predator on prey and the effect of prey on predator. The second one is being in the periodic environment. In this study, by using these two issues, the necessary and sufficient conditions to have globally attractive or globally asymptotically stable solution for some different types of predator-prey systems are found.

First, let us give some information about the meaning of functional response and the types of functional responses. As it is remembered above, functional responses show how and how much predator gets benefit from the prey and how and how much prey is affected by predator. There are many types of functional responses. Some of them are Holling type, Beddington-DeAngelis type, ratio type, monotype, semiratio type, and so on. The following studies are some examples about predator-prey models with Holling type functional response: . Studies that are about Beddington-DeAngelis type functional response are . Predator-prey systems with other types of functional responses are studied in . In this study, the main aim is to formulize the functional responses in a general form. In other words, the essential object for this study is to be able to express different types of functional responses in one functional response. Therefore, the dynamical properties that are valid for this system become also valid and useful for the other types.

The second issue in that study is to be in a periodic environment. In nature, periodicity can be seen in many different circumstances. For example, many animals ovulate periodically or many insects have periodic life cycle. Therefore, the analysis of the predator-prey dynamic system in a periodic environment is very significant. On the other hand, in population growth model, the significantly studied problem is the stability and global existence of a positive periodic solution in that system. In an autonomous model, the globally stable equilibrium point is the same as the notion globally asymptotically stable positive periodic solution in a nonautonomous system. Hence, the main problem in this study is to determine under which conditions globally attractive positive periodic solution is attained for the systems with different functional responses that are obtained from the system with generalized functional response. Additionally the importance of this issue can be seen from previous studies, since this subject is investigated in those ones. Some of the examples can be given as follows: [3, 1020].

In study , some kinds of impulsive predator-prey systems have been investigated on time scales. Nevertheless, the functional response that affects predator is not used in . Only its effect on prey is considered. Hence, the results of that paper only related to generalized type of semiratio dependent functional responses. However, in this study, the main aim is to obtain a model with a generalized functional response both on prey and on predator. Because of the above reason, this study has important contributions.

On the other hand, the systems with Holling type I and II functional responses which are obtained from the system with generalized functional response are also studied. For the system with Holling type I functional responses, in paper , an if and only if condition has been found to obtain the permanence of the given system. However, by using the main result of our study, an if and only if condition can be given for the globally asymptotically stable periodic solution of the considered system. Also in the study of , in the predator part, to bound the solution of predator from above an extra term has been used, but we can get rid of that extra term in this study. These are some points that show the importance of our study.

As a result, the primary objective of that study is to generalize the functional responses that act both on predator and on prey. Additionally, the second aim is to find the if and only if condition for the globally asymptotically stable periodic solution of the considered systems with some different types of functional responses.

2. Preliminaries

As preliminary, we use Definition 1, Lemma 1, Theorem 1, and necessary information that is needed for Theorem 1 from .

3. Main Result

The following is the main equation for this study:(1)x~t=atx~t-btx~2t-Φ1t,x~t,y~ty~t,y~t=-dty~t+Φ2t,x~t,y~tx~t.

In system (1), Φ1 satisfies the following: Φ1(t,0,y)=0, is continuous and w-periodic, and satisfies the following inequality:(2)γ0txm1t++γm1-1txtΦ1t,xt,ytα0txm1t++αm1-1txt.Assumptions on Φ2 are as follows: Φ2(t,x,0)=0, is continuous and w-periodic, and satisfies the following inequality:(3)Φ2t,xt,ytβ0tym2++βm2-1tyt.For the above inequalities, αi, βj, and γj are positive, w-periodic coefficient functions for i=1,2,,m1-1 and j=1,2,,m2-1. Also, Φ1(t,x(t),y(t)) and Φ2(t,x(t),y(t))>0 if x(t) and y(t)>0. In addition to these conditions, Φ2(t,expx,exp(y))-dΦ2(t,expx,exp(y))/dy>0 and a(t), b(t), and d(t) are positive and w-periodic coefficient functions.

a(t)x~(t)-b(t)x~2(t) is the specific growth rate of the prey in the absence of predator;

d(t) is the death rate of predator;

-Φ1(t,x~(t),y~(t))y~(t) is generalized effect of predator on prey;

Φ2(t,x~(t),y~(t))x~(t) is the generalized effect of prey on predator.

Assume that y~(t)=exp(y(t)) and x~(t)=exp(x(t)). Therefore, system (1) becomes equal to the following system:(4)xt=at-btexpxt-Φ1t,expxt,expytexpyt-xt,yt=-dt+Φ2t,expxt,expytexpxt-yt.

Theorem 1.

In system (4), all of the coefficient functions satisfy the above conditions. System (4) has at least one w-periodic solution if and only if 0wa(t)dt>0 and 0wd(t)dt>0.

Proof.

When there is at least one w-periodic solution for system (4), then it is apparent that 0wa(t)dt>0 and 0wd(t)dt>0. Since (5)0watdt=0wbtexpxt+Φ1t,expxt,expytexpyt-xtdt>0,0wdtdt=0wΦ2t,expxt,expytexpxt-ytdt>0,now, for the converse part, assume that 0wa(t)dt,  0wd(t)dt>0 and try to find the fact that system (4) has at least one w-periodic solution.

Let X, Y be the normed vector spaces and let the operators L, N be the same as they are defined in the proof of Theorem  2 in . By using the similar proof techniques of Theorem  2 from , then for any open bounded set ΩX, we can obtain that N is L-compact on Ω¯.

The below system is investigated by using the application of Theorem 1 from :(6)xt=λat-btexpxt-Φ1t,expxt,expytexpyt-xt,yt=λ-dt+Φ2t,expxt,expytexpxt-yt.

Let xyX solve system (6). If system (6) is integrated from 0 to w, then(7)0watdt=0wbtexpxt+Φ1t,expxt,expytexpyt-xtdt,0wdtdt=0wΦ2t,expxt,expytexpxt-ytdt.

By using the first equations of (6) and (7), we get(8)0wxtdtλ20watdtS1,where S120wa(t)dt.

By using the second equations of (6) and (7), we have(9)0wytdtλ20wdtdtS2;here S220wd(t)dt.

On compact sets, continuous functions attain its maximum and minimum. Therefore, there exist ψi,θi,  i=1,2, for xyX such that(10)xθ1=mint0,wxt,xψ1=maxt0,wxt,(11)yθ2=mint0,wyt,yψ2=maxt0,wyt.The following is obtained by using the first equations of (7) and (11):(12)xθ1<l~1,where  l~1ln0watdt0wbtdt.Using the first inequality in Lemma 1 from  and inequality (8) we have(13)xtxθ1+0wxtdtxθ1+20watdt<P1l~1+S1.

Using the first equation of (7), the below result is obtained: (14)0watdt0wγ0texpxtm1++γm1-1texpxtexpyt-xtdt0wγm1-1texpyt-xtdtexpyθ20wγm1-1tdtexpyθ20wγm1-1tdt.Therefore,(15)expyθ20wdtdt0wγm1-1tdtl~2.Using the first inequality in Lemma 1 from  and inequality (9), we get(16)ytyθ2+0wytdtyθ2+20wdtdt<P2l~2-S2.From the second equation of (7), we have (17)0wdtdt0wβ0texpytm2++βm2-1expytexpxt-ytdt0wβ0texpytm2-1++βm2-1expxtdtexpxψ10wβ0texpP2m2-1++βm2-1dt.Hence,(18)xψ1ln0wdtdt0wβ0texpP2m2-1++βm2-1dtl~3.

Using the second inequality in Lemma 1 from  and inequality (8), we have(19)xtxψ1+0wxtdtxψ1+20watdt>P3l~3+S1.

Again by using the first equations of (7) and (13), we get (20)0watdt0wbtexpxt+α0texpxtm1++αm1-1texpxtexpyt-xtdt0wα0texpxtm1-1++αm1-1t+btexpytdtexpyψ20wα0texpxtm1-1++αm1-1t+btdtexpyψ20wα0texpP1m1-1++αm1-1t+btdt.Therefore the following is obtained:(21)expyψ20watdt0wα0texpP1m1-1++αm1-1t+btdtl~4.

Using the second inequality in Lemma 1 from  and (9), we get(22)ytyψ2-0wytdtyψ2-20wdtdt>P4l~4-S2.

By (13) and (19), we have supt[0,w]xtC1max{P1,P3} and by (16) and (22), we have supt[0,w]ytC2max{|P2|,|P4|}. Here, C1 and C2 are independent from λ. Let S=C1+C2+1. In addition to these,(23)maxt0,wxy<S.

Suppose that Ω={xyX:xy<S} verifies the first requirement of Theorem 1 from . When xyKerLΩ, then we have xy=S, where S is a constant. Thus, (24)QNxy0was-bsexpx-Φ1s,expx,expyexpy-xds0w-ds+Φ2t,expx,expyexpx-yds.

The operator J:ImVKerL is taken as the identity operator and the homotopy is defined as (25)Hν=νJQN+1-νG,where (26)Gxy=0was-bsexpxds0wds-Φ2t,expx,expyexpx-yds.

Let DJG be the determinant of the Jacobian of G. We know that xyKerL; then the Jacobian of G is (27)-ex0wbsds00wex-yΦ2t,expx,expy+dΦ2t,expx,expydxds-0wex-yΦ2t,expx,expy-dΦ2t,expx,expydyds.

Here signDJG is always positive and this is obtained by the assumption of the theorem. Hence(28)degJN,ΩKerL,0=degG,ΩKerL,0=xyG-100signDJGxy0.

Thus, all the conditions of Theorem 1 from  are satisfied. Therefore, system (1) has at least one positive w-periodic solution.

3.1. Predator-Prey Systems with Different Types of Functional Responses 3.1.1. Predator-Prey System with Holling Type Functional Response

One of the applications of the above result for Theorem 1 is the Holling type functional response from .

System  (3.13) from  for continuous case is (29)x~t=atx~t-btx~2t-ctx~ty~t1+mtx~t,y~t=-dty~t+fty~tx~t1+mtx~t.When x~(t)=exp(x(t)) and y~(t)=exp(y(t)), we obtain the transformed form of this system as (30)xt=at-btexpxt-ctexpyt1+mtexpxt,yt=-dt+ftexpxt1+mtexpxt.After assuming the system has positive coefficient functions, it is obvious that we can take (31)Φ1t,xt,yt=ctxt1+mtxt,Φ2t,xt,yt=ftyt1+mtxt.It is also apparent that Φ1(t,x(t),y(t))c(t)x(t) and Φ2(t,x(t),y(t))f(t)y(t). Here if prey does not go to extinction, then there exists kN such that for sufficiently large k, Φ1(t,x(t),y(t))(1/k)c(t)x(t), since c(t)x(t)/(1+mtxt)>0. Therefore, Theorem 1 has equal statement with the following remark for the system with Holling type functional response.

Remark 2.

For system  (3.13) from  for continuous case, there exists at least one w-periodic solution if and only if the prey does not become extinct.

Remark 3.

Here by using proof techniques of Lemma 3 from , it can be seen that if predator does not become extinct, then prey also does not become extinct. Therefore, Remark 2 becomes as follows: system  (3.13) from  for continuous case has w-periodic solution if and only if predator does not become extinct.

Lemma 4.

For system  (3.13) from  for continuous case, predator becomes extinct if and only if (32)0w-dt+ftxt1+mtxt0,where x(t)=1-exp-0wasds/0wbt-sexp(-0sa(t-τ)dτ)ds. Here x(t) is the globally attractive unique solution of the system (33)xt=atxt-btx2t.

Proof.

Assume that predator goes to extinction. The first equation of the predator-prey system with Holling type functional response is(34)x~t=atx~t-btx~2t-ctx~ty~t1+mtx~t.

Since as t tends to infinity, predator or y~(t)=exp(y(t)) tends to zero. Hence, equality (34) tends to the following equation as t tends to infinity: (35)x~t=atx~t-btx~2t.This means that as t tends to infinity the solution of x~(t) tends to x(t).

Since predator goes to extinction, when we take exp(y(t))=y~(t), then from the second equation of this transformed version of the system with Holling type functional response the following inequality is obtained as(36)yt=0w-dt+ftexpxt1+mtexpxt0.Since t tends to infinity, the solution of x~(t)=exp(x(t)) tends to x(t). Thus, equality (36) tends to (37)yt=0w-dt+ftxt1+mtxt.Since 0w-d(t)+ftexp(x(t))/(1+mtexpxt)0, then also 0w-d(t)+f(t)x(t)/(1+mtxt)0.

For the converse, let us assume that 0w-d(t)+f(t)x(t)/(1+mtxt)0. Consider the first equation of the system with Holling type functional response, then we obtain (38)x~t=atx~t-btx~2t-ctx~ty~t1+mtx~tatx~t-btx~2t.About the solution of prey, the result exp(x(t))=x~(t)x(t) is obtained by the comparison theorem for ODEs.

Since 0w-d(t)+f(t)x(t)/(1+mtxt)0, then y(t)0, which means predator goes to extinction. Hence the proof follows.

Lemma 5.

In system  (3.13) from  take T=R; then this system has at least one w-periodic solution if and only if(39)0w-dt+ftxt1+mtxt>0,where x(t)=(1-exp-0wasds)/0wbt-sexp(-0sa(t-τ)dτ)ds.

Proof.

By using Lemma 4 and Remarks 2 and 3, one can prove Lemma 5.

The permanence definition is taken from .

Corollary 6.

For system  (3.13) from  for continuous case, the solution is permanent if and only if (39) is satisfied.

Proof.

Let us take (39) as satisfied. Therefore, from Lemma 4, predator does not become extinct and from Remark 3, prey does not become extinct also which means that both x~(t) and y~(t) are bounded from below with a positive number. From  (Proposition  3.1), the solution of the prey x~(t) is bounded from above. Assume that (39) is satisfied and y~(t) is not bounded from above which means that as t tends to infinity y~(t) also tends to infinity. Consider the first equation of transformed version of the system with Holling type functional response, which is (40)xt=at-btexpxt-ctexpyt1+mtexpxt.

Because of the assumption, as t tends to infinity y~(t)=exp(y(t)) also tends to infinity; then also y(t) tends to infinity. Since we have found above that x~(t)=exp(x(t)) is bounded from above and below with positive constants, then x(t) is bounded from above and below with constants from real numbers. Hence, for sufficiently large T, x(t) tends to - as t tends to infinity. Therefore, x~(t) tends to 0 as t tends to infinity. This is a contradiction with the boundedness of x~(t) from below with a positive constant. Therefore, y~(t) is bounded from above. Hence the solution of the considered system is permanent.

For the converse, let us assume that the system is permanent. Then prey and predator do not go to extinction; then by Remark 2 the considered system has at least one w-periodic solution. Thus, by using Lemma 5 obviously, the system satisfies (39).

Theorem 7.

Assume that (39) holds true. Then, the w-periodic solution of system  (3.13) from  for continuous case is globally attractive or globally asymptotically stable.

Proof.

Proof is similar with the proof of Lemma  5 in .

Corollary 8.

Consider the same conditions for the coefficient functions of system  (3.13) from . There exists w-periodic globally attractive solution for this system if and only if (39) is satisfied.

Proof.

The result is obtained from Theorem 7 and Lemma 5.

3.1.2. Predator-Prey System with Holling Type II Functional Response

As a second case, consider the following predator-prey system with Holling type II functional response which is the same as system  (3.14) from  for continuous case:(41)x~t=atx~t-btx~2t-ctx~2ty~t1+mtx~2t,y~t=-dty~t+fty~tx~2t1+mtx~2t.

When y~(t)=exp(y(t)) and x~(t)=exp(x(t)), the transformed form of this system is gotten as(42)xt=at-btexpxt-ctexpyt+xt1+mtexp2xt,yt=-dt+ftexp2xt1+mtexp2xt.Assume that all the coefficient functions are positive for the above systems. Then (43)Φ1t,xt,yt=ctx2t1+mtx2t,Φ2t,xt,yt=ftxtyt1+mtx2t,Φ1(t,x(t),y(t))c(t)x2(t) and Φ2(t,x(t),y(t))f(t)Mxy(t), where Mx is the maximum of the solution of system (41) for prey which can be found from Proposition  3.1 in . Here if prey does not go to extinction, then there exists kN such that, for sufficiently large k, Φ1(t,x(t),y(t))(1/k)c(t)x2(t), since c(t)x2(t)/(1+mtxt)>0. Therefore, Theorem 1 has equal statement with the following remark for the systems with Holling type II functional response.

Remark 9.

For continuous case of system  (3.14) from  it has at least one w-periodic solution if and only if the prey does not go to extinction.

By using similar techniques of Lemma 3 from , it can be found that if predator does not go to extinction, then prey also does not go to extinction. Therefore, the statement of Remark 9 is the same as the following one: system  (3.14) from  for continuous case has at least one w-periodic solution if and only if predator does not go to extinction.

Lemma 10.

For system  (3.14) from , take T=R. Then predator becomes extinct if and only if (44)0w-dt+ftx2t1+mtx2t0,where x(t)=(1-exp-0wasds)/0wbt-sexp(-0sa(t-τ)dτ)ds. Here x(t) is the unique globally attractive solution of the system (45)xt=atxt-btx2t.

Proof.

Proof is similar to the proof Lemma 4.

Lemma 11.

In system  (3.14) from  when it is taken as T=R, the system has at least one w-periodic solution if and only if(46)0w-dt+ftx2t1+mtx2t>0,where x(t)=(1-exp-0wasds)/0wbt-sexp(-0sa(t-τ)dτ)ds.

Proof.

This lemma can be proven by Lemma 10 and Remark 9.

Corollary 12.

In system  (3.14) from  when it is taken as T=R, the solution of the considered system is permanent if and only if (46) is satisfied.

Proof.

The proof is similar to the proof of Corollary 6.

Theorem 13.

Assume that (46) holds true. Hence, the w-periodic solution of system  (3.14) from  for continuous case is globally attractive or globally asymptotically stable.

Proof.

This has a similar proof with the proof of Lemma  5 in .

Corollary 14.

Assume that the same conditions for the coefficient functions of system  (3.14) from  hold. There exists w-periodic globally attractive solution for this system if and only if (46) is satisfied.

Proof.

By Theorem 13 and Lemma 11, one can get the desired result.

3.1.3. Predator-Prey System with Beddington-DeAngelis Type Functional Response

Consider the following system which is taken from :(47)x~t=atx~t-btx~2t-cty~tx~tαt+βtx~t+mty~t,y~t=-dty~t+ftx~ty~tαt+βtx~t+mty~t.

If we take exp(x(t))=x~(t), then we have the following transformed system:(48)xt=at-btexpxt-ctexpytαt+βtexpxt+mtexpyt,yt=-dt+ftexpxtαt+βtexpxt+mtexpyt.For the above systems, all the coefficient functions are positive. It is obvious that when we take (49)Φ1t,xt,yt=ctxtαt+βtxt+mtyt,Φ2t,xt,yt=ftytαt+βtxt+mtyt,Φ1(t,x(t),y(t))(ct/αt)x(t) and Φ2(t,x(t),y(t))(ft/αt)y(t). Here if prey does not go to extinction, then there exists kN such that, for sufficiently large k, Φ1(t,x(t),y(t))(1/k)(ct/αt)x(t), since c(t)x(t)/(αt+βtxt+mtyt)>0. Hence, Theorem 1 is true for system (47). These findings are the same as the ones in .

4. Examples Example 1.

(50) x t = 3 - cos t + 2 exp x t - 2 exp y t exp x t + 1 , y t = - 0.1 sin t + 0.5 + exp x t exp x t + 1 .

By doing some simple calculations, it can be seen that x>1; therefore (39) is satisfied for Example 1, since (51)02π-0.1sint+0.5+xx+1>0.Figure 1 supports this result.

Even if we have changed the initial conditions, after a while still we obtain the same solution which shows the global attractivity of the solutions. Figure 2 supports this result.

x ( 0 ) = 0.2 and y(0)=0.4.

x ( 0 ) = 2 and y(0)=1.

Example 2.

(52) x t = 3 - cos t + 2 exp x t - 2 exp y t exp x t + 1 , y t = - 0.1 sin t + 4 + exp x t exp x t + 1 .

Example 2 does not satisfy inequality (39). This is easily seen by doing some simple calculations. Hence predator goes to extinction which satisfies Lemma 4. This result is supported by Figure 3.

x ( 0 ) = 0.3 and y(0)=0.5.

Example 3.

(53) x t = 3 - cos t + 2 exp x t - 2 exp y t + x t exp 2 x t + 1 , y t = - 0.1 sin t + 1 + 3 exp 2 x t exp 2 x t + 1 .

By doing some simple calculations it can be seen that x>1; therefore (46) is satisfied by Example 3, since (54)02π-0.1sint+1+3x2x2+1>0.5.This is supported by Figure 4.

Even if we have changed the initial conditions, after a while, we still obtain the same solution which shows the global attractivity of the solutions. Figure 5 supports this.

x ( 0 ) = 0.3 and y(0)=0.5.

x ( 0 ) = 1 and y(0)=1.

Example 4.

(55) x t = 3 - cos t + 2 exp x t - 2 exp y t + x t exp 2 x t + 1 , y t = - 0.1 sin t + 6 + 3 exp 2 x t exp 2 x t + 1 .

Example 4 does not satisfy inequality (46) and this can be easily seen by doing some simple calculations. Therefore, predator goes to extinction which satisfies Lemma 10 and this is supported by Figure 6.

For the examples of the system with Beddington-DeAngelis functional response, look at the study .

x ( 0 ) = 0.3 and y(0)=0.5.

5. Discussion

In that study, two important analytical results are found. First, for the periodic solution of the system with generalized functional response, the if and only if condition is able to be found. Therefore, we are able to extend the study in  for continuous case. The second one is to be able to find an if and only if condition for the globally periodic solution of the predator-prey models with Holling type I and II functional responses. Additionally, the results of that paper is consistent with the findings of the paper  when the Beddington-DeAngelis type functional response is considered as an application.

Hence, the allover work in this study is important, since by using the results of that paper, the if and only if condition for at least one periodic solution and the globally attractive periodic solution can be found and these results can be generalized in many different types of functional responses.

The suggested problem for the future works is to find the if and only if condition for the globally attractive periodic solution of the discrete predator-prey dynamic systems. Semigroup theory has been used in the analysis of the system when it has globally attractive periodic solution of the continuous system. For the discrete case, for further studies, the result that is related to the global attractivity of the system is another open problem.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this article.

Bohner M. Fan M. Zhang J. Existence of periodic solutions in predator-prey and competition dynamic systems Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal 2006 7 5 1193 1204 10.1016/j.nonrwa.2005.11.002 MR2260908 2-s2.0-33745116193 Wang W. Shen J. Nieto J. J. Permanence and periodic solution of predator-prey system with holling type functional response and impulses Discrete Dynamics in Nature and Society 2007 2007 15 81756 10.1155/2007/81756 MR2346521 2-s2.0-36949038244 Xu R. Chaplain M. A. J. Davidson F. A. Periodic solutions for a predator-prey model with Holling-type functional response and time delays Applied Mathematics and Computation 2005 161 2 637 654 10.1016/j.amc.2003.12.054 MR2112430 2-s2.0-10044278294 Pelen N. N. Güvenilir A. F. Kaymakçalan B. Necessary and sufficient condition for existence of periodic solutions of predator-prey dynamic systems with Beddington-DeAngelis-type functional response Advances in Difference Equations 2016 2016 1, article 15 19 10.1186/s13662-016-0747-0 2-s2.0-84955473568 Pelen N. N. Güvenilir A. F. Kaymakçalan B. Behavior of the solutions for predator-prey dynamic systems with Beddington-DeAngelis type functional response on periodic time scales in shifts Abstract and Applied Analysis 2016 2016 9 1463043 10.1155/2016/1463043 MR3457394 2-s2.0-84957989578 Guvenilir A. F. Kaymakalan B. Pelen N. Impulsive predator-prey dynamic systems with Beddington-DeAngelis type functional response on the unification of discrete and continuous systems Applied Mathematics 2015 6 9 1649 1664 10.4236/am.2015.69147 Xu C. Liao M. Existence of periodic solutions in a discrete predator-prey system with Beddington-DeAngelis functional responses International Journal of Mathematics and Mathematical Sciences 2011 2011 18 970763 10.1155/2011/970763 MR2861137 2-s2.0-84855523894 Yang L. Yang J. Zhog Q. Periodic solutions for a predator-prey model with Beddington-DeAngelis type functional response on time scales General Mathematics Notes 2011 3 1 46 54 Zhang J. Wang J. Periodic solutions for discrete predator-prey systems with the Beddington-DeAngelis functional response Applied Mathematics Letters 2006 19 12 1361 1366 10.1016/j.aml.2006.02.004 MR2264191 2-s2.0-33748305640 Fan M. Wang K. Periodicity in a delayed ratio-dependent predator-prey system Journal of Mathematical Analysis and Applications 2001 262 1 179 190 10.1006/jmaa.2001.7555 MR1857221 Fan M. Wang Q. Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems Discrete and Continuous Dynamical Systems. Series B 2004 4 3 563 574 10.3934/dcdsb.2004.4.789 MR2073960 Huo H.-F. Periodic solutions for a semi-ratio-dependent predator-prey system with functional responses Applied Mathematics Letters 2005 18 3 313 320 10.1016/j.aml.2004.07.021 MR2121442 2-s2.0-13844280309 Wang Q. Fan M. Wang K. Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses Journal of Mathematical Analysis and Applications 2003 278 2 443 471 10.1016/s0022-247x(02)00718-7 MR1974018 2-s2.0-0038066660 Chen F. Permanence and global stability of nonautonomous Lotka-Volterra system with predator-prey and deviating arguments Applied Mathematics and Computation 2006 173 2 1082 1100 10.1016/j.amc.2005.04.035 MR2207997 2-s2.0-32644439511 Cui J. Takeuchi Y. Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response Journal of Mathematical Analysis and Applications 2006 317 2 464 474 10.1016/j.jmaa.2005.10.011 MR2209573 2-s2.0-33644590203 Fan M. Agarwal S. Periodic solutions for a class of discrete time competition systems Nonlinear Studies 2002 9 3 249 261 MR1918904 Fan M. Kuang Y. Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response Journal of Mathematical Analysis and Applications 2004 295 1 15 39 10.1016/j.jmaa.2004.02.038 MR2064405 2-s2.0-2942641510 Fan M. Wang K. Global periodic solutions of a generalized n-species Gilpin-Ayala competition model Computers & Mathematics with Applications 2000 40 10-11 1141 1151 10.1016/s0898-1221(00)00228-5 MR1784658 2-s2.0-0034324869 Fang Q. Li X. Cao M. Dynamics of a discrete predator-prey system with Beddington-DeAngelis function response Applied Mathematics 2012 3 4 389 394 10.4236/am.2012.34060 MR2917037 Li H. Takeuchi Y. Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response Journal of Mathematical Analysis and Applications 2011 374 2 644 654 10.1016/j.jmaa.2010.08.029 MR2729250 2-s2.0-77957768633 Liu X. Liu X. Necessary and sufficient conditions for the existence of periodic solutions in a predator-prey model on time scales Electronic Journal of Differential Equations 2012 2012 199 1 13 Zbl1290.34056 Cui J. Permanence of predator-prey system with periodic coefficients Mathematical and Computer Modelling 2005 42 1-2 87 98 10.1016/j.mcm.2005.03.001 MR2162389 2-s2.0-26844468189