We study a general Gause-type predator-prey model with monotonic functional response under Dirichlet boundary condition. Necessary and sufficient conditions for the existence and nonexistence of positive solutions for this system are obtained by means of the fixed point index theory. In addition, the local and global bifurcations from a semitrivial state are also investigated on the basis of bifurcation theory. The results indicate diffusion, and functional response does help to create stationary pattern.

In this paper, we are interested in the following semilinear elliptic system with monotonic functional response under Dirichlet boundary condition:

In this work, we aim to understand the influence of diffusion and functional response on pattern formation, that is, the positive solutions of (

The rest of this paper is organized as follows. In Section

In order to give the main results and complete the corresponding proofs, we need to introduce some necessary notations and theorems as the following.

For each

In order to calculate the indexes at the trivial and semitrivial states by means of the fixed point index theory, we also need to introduce the following theorem.

Assume

It is easy to see that the corresponding conclusions in Theorem

From Theorem

Let

if

if

if

Consider the following equation:

Assume that the function

Let

It is easy to see that if the function

Now, we introduce the fixed point index theory which plays an important role in finding the sufficient conditions for the existence of positive solutions of model (

Let

Assume that

If

If

Finally, we introduce a result about global bifurcation, which was introduced by López-Gómez and Molina-Meyer in [

Let

The solutions of (

Define the parity mapping

Suppose that

Moreover, if

When we are working in a product-ordered Banach space, the conditions (

At first, we introduce the following lemma which gives the necessary condition for (

If problem (

Assume

In the rest of this section, we shall prove that the necessary conditions in Lemma

Assume

It is obvious that

Now, we introduce the following notations:

where

Define a positive and compact operator

(

(

(

At first, we shall calculate the topological degree of the operator

For

Assume that

Now, we need to calculate the fixed point index of the operator

Assume that

if

if

(i) Observe

We claim that

From Remark

Now, we can prove that

If

If

From Lemmas

Problem (

Let

Problem (

From Corollary

In this subsection, we will employ the local bifurcation theory [

Assume that

Let us introduce the change of variable

Introduce an operator

If

For such a solution, the first equation enables us to obtain

In order to investigate the bifurcation direction from

According to the theory of Rabinowitz [

In this subsection, basing on the results in Theorem

Assume that

Let

Subsequently, for every

In order to complete the proof of Theorem

Since

From the definition of the operator

It is easy to see that

This work was supported by the Fundamental Research Funds for the Central Universities (no. XDJK2009C152).