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A plateau pika model with spatial cross-diffusion is investigated. By analyzing the corresponding characteristic equations, the local stability of an coexistence steady state is discussed when

Plateau pika (

During the recent years, the various methods are applied to manage the number of plateau pikas. In the first stage, since people are lacking enough understanding about the plateau pika, they directly poisoned the plateau pike with botulin of models C and D. The people considered the plateau pika as rodent; in fact, pikas belong to lagomorphs and not to rodents. From the first paragraph, plateau pike is keystone species, whose elimination or major decimation from an ecosystem would have a greater average effect on other species’ populations or ecosystem processes. Hence, plateau pika contributes to maintaining the biodiversity in Qinghai-Tibet plateau [

Animal populations may not be distributed randomly in space but exhibit spatial patterns, and plateau pika is not exception. Plateau pika spatial distribution in the plateau is directly affected by individual interactions and interspecific interactions. Plateau pika dispersing is a natural phenomenon. Pikas from the different plateaus often contact and move towards favorable habitats due to climate, food, predators, and so on. This phenomena can employ random diffusion combined with directed movement upward along environmental gradients. Many interesting papers investigating species diffusion have been discussed by many authors. In [

This paper is organized as follows. Section

For the single species model under contraceptive control and lethal control, it is mentioned in the introduction [

Here,

When

In this paper, we are concerned with the effect of the cross-diffusion under the contraceptive control and lethal control. The transmission dynamics of infection are governed by the following differential equations:

For the convenience to study the dynamics of (

Assume that

Suppose that

In this section, first we cite Lemma

Let

By a direct calculation, we have

Consider

In this case, it is easy to get the characteristic equation of the coexistence steady state

If

Consider

From the computation and notations, the trace of

It is easy to know that, if

Assume

From the definition of

When

When

In this section, we will be interested primarily in steady state of (

In the following, we will discuss the upper bounds and the lower bounds by using Lemma

Assume that

For using the Lemma

Let

From Theorem

Assume that

Since

Now we need to estimate the positive lower bound of

Taking the limit and integrating the second equation of (

By Theorems

In order to discuss the existence of nonconstant positive solution of (

Assume

Let

In the following, we consider that the cross-diffusion induces the nonconstant solution for (

In order to use Leray-Schauder degree, we must revise system (

If

Recall that

In the following, we consider how the cross-diffusion coefficient

Assume

Next we will discuss the existence of nonconstant positive solution of (

Assume that the other parameters in (

Due to Lemma

For

It is easy to check that

By the definition of

In addition,

We have formulated a plateau pika model with cross-diffusion. By Lemma

We can illustrate the theoretical results through two simulatinon examples. Set

Numerical simulation illustrates our result (see Figure

Plateau pika coexistence steady state

Plateau pika coexistence steady state

For the management of the application, larger

Unfortunately, we are unable yet to study global stability of coexistence steady state

The authors declare that they have no conflict of interests.

This study is supported by the NSF of China (61203228, 61275120, and 11371313), Mathematical Tianyuan Foundation (11226258), the Young Sciences Foundation of Shanxi (2011021001-1), the Natural Science Foundation of Shanxi Province (2013011002-5), program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi, and the “131” talents.