In this paper, we consider two kinds of time-delay slow-fast modified Leslie-Gower models. For the first system, we prove the existence and uniqueness of relaxation oscillation cycle through the geometric singular perturbation theory and entry-exit function. For the second system, we put forward a conjecture that the relaxation oscillation of the system is unique. Numerical simulation also verifies our results for the systems.

In 2003, Alaoui and Okiye proposed the next modified Leslie–Gower model:

Consider the rescaling,

Substitute (

Consider the case where

The geometric singular perturbation theory is mathematical rigorous that can analyze dynamics of some slow-fast systems. The study of invariant manifolds in Fenichel’s theory [

Recently, Wang and Zhang proved the existence and uniqueness of relaxation oscillation cycle of the slow-fast modified Leslie-Gower model [

The rest of the paper is organized as follows: Section

We are going to consider a slow-fast vector field in the form of

For system (

(a) For

Consider a system with time delays

Using the Taylor formula,

Substitute (

For system (

Moreover, if

We consider the bifurcation of system (

Then, the characteristic equation of system (

For equation (

The condition of equation (

Consider

According to the equation

Taking

The proof is completed.

Let

If

The condition of equation (

The proof is completed.

Denote

For system (

Consider the limit systems of system (

Consider the slow time scale

We restrict the parameters

Under this restriction,

Illustration for the relaxation oscillations of system (

In fact, when the parameter is limited to

For system (

Considering the system (

So that

Furthermore

Then, we combine (

The proof is completed.

Let us define

For system (

By Fenichel’s theory, the critical submanifold

Let

Then,

Now, we analyze the properties of these two maps

Therefore, according to the results of (a) and (b), we obtain that the transition map

According to Fenichel’s theory and Theorem

So, we can get that the relaxation oscillation cycle

The theorem is proved.

First of all, we provide an example to illustrate Theorem

Select the parameter values

The orbit is shown in red in phase space

Time series with the black dashed curve (prey) and red full curve (predator)

Numerical simulation of system (

In addition, we compare the phase portrait and time series of system (

Time series for the relaxation oscillation of the predator with the red dashed curve (system (

Time series for the relaxation oscillation of the prey with the blue dashed curve (system (

Phase space: the dashed blue curve shows the orbit of system (9) and the red full curve shows the orbit of system (

Time series of system (

In this section, we consider time delays

System (

In order to get the conditions for Hopf bifurcation, we reach the linearized system of system (

Substituting

Denote

For equation (

Let

Separating the real and imaginary parts of equation (

So that

According to

Since

We can conclude that if

The proof is completed.

Denote

(transversality condition) If

Differentiating both sides of equation (

So

To sum up, we get

The proof is completed.

For system (

If

The theorem is proved.

If

In order to get rid of the delay

When

Consider the case where

Then, we arrive at the critical set:

Since

Thus, we reach the attractive part and the repulsive part of

The attractive part and the repulsive part of

The attractive part and the repulsive part of

In this section, we combine the result of Subsection

If

We show some numerical simulations in this part to verify our theoretical results.

For system (

Choose

Choose

For system (

Numerical simulation of system (

In this paper, we mainly study two kinds of time-delay slow-fast modified Leslie-Gower models. For the first system, through the geometric singular perturbation theory and entry-exit function, we prove the existence and uniqueness of relaxation oscillation cycle. For another system, when

All data, models, and code generated or used during the study are included within the article.

The authors declare that they have no conflicts of interest.

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grant no.11572288 and the Natural Science Foundation of Zhejiang through grant no.LY20A020003.