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.

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: [

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: [

In study [

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 [

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.

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

The following is the main equation for this study:

In system (

Assume that

In system (

When there is at least one

Let

The below system is investigated by using the application of Theorem 1 from [

Let

By using the first equations of (

By using the second equations of (

On compact sets, continuous functions attain its maximum and minimum. Therefore, there exist

Using the first equation of (

Using the second inequality in Lemma 1 from [

Again by using the first equations of (

Using the second inequality in Lemma 1 from [

By (

Suppose that

The operator

Let

Here

Thus, all the conditions of Theorem 1 from [

One of the applications of the above result for Theorem

System (3.13) from [

For system (3.13) from [

Here by using proof techniques of Lemma 3 from [

For system (3.13) from [

Assume that predator goes to extinction. The first equation of the predator-prey system with Holling type functional response is

Since as

Since predator goes to extinction, when we take

For the converse, let us assume that

Since

In system (3.13) from [

By using Lemma

The permanence definition is taken from [

For system (3.13) from [

Let us take (

Because of the assumption, as

For the converse, let us assume that the system is permanent. Then prey and predator do not go to extinction; then by Remark

Assume that (

Proof is similar with the proof of Lemma 5 in [

Consider the same conditions for the coefficient functions of system (3.13) from [

The result is obtained from Theorem

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 [

When

For continuous case of system (3.14) from [

By using similar techniques of Lemma 3 from [

For system (3.14) from [

Proof is similar to the proof Lemma

In system (3.14) from [

This lemma can be proven by Lemma

In system (3.14) from [

The proof is similar to the proof of Corollary

Assume that (

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

Assume that the same conditions for the coefficient functions of system (3.14) from [

By Theorem

Consider the following system which is taken from [

If we take

By doing some simple calculations, it can be seen that

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

Example

By doing some simple calculations it can be seen that

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

Example

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

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 [

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.

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