A special predator-prey system is investigated in which the prey population exhibits herd behavior in order to provide a self-defense against predators, while the predator is intermediate and its population shows individualistic behavior. Considering the fact that there always exists a time delay in the conversion of the biomass of prey to that of predator in this system, we obtain a delayed predator-prey model with square root functional response and quadratic mortality. For this model, we mainly investigate the stability of positive equilibrium and the existence of Hopf bifurcation by choosing the time delay as a bifurcation parameter.

During the last few decades, there has been great interest in the construction and study of models for the population dynamics of predator-prey systems. The classical predator-prey model can be written in the generalized form (see [

For system (

Recently, one kind of predator-prey system with herd behavior is considered by some researchers [

Moreover, there are some papers concerned with the effects of the form of mortality terms for the dynamics [

On the other hand, for most of the natural ecosystems, every species does not respond instantaneously to changes in the environment or the interactions with other species within the community. Thus models with delay are much more realistic [

Motivated by the above, in the present paper, we are intended to consider a predator-prey system in which the prey species exhibits herd behavior, the predator species is intermediate, and there exists a time delay in the convention of the biomass of prey to that of predator.

This paper is organized as follows. We first introduce our working system in the next section. In Section

Braza [

Following References [

By the same way of [

It is easy to verify that system (

Due to biological interpretation of the system, we only consider the positive equilibrium. In this section, by choosing

To study the local stability of the positive equilibrium

The linear equations corresponding to (

When there is no delay, that is,

Assume

Now for

Assume

If

If

Next, under the condition of Lemma

Assume

Summarizing the above three lemmas, we can obtain the following theorem on stability and Hopf bifurcation of system (

Assume

If

If

In the previous section, we have obtained the conditions under which a family of periodic solutions bifurcate from the positive equilibrium

Assume that system (

The direction of the Hopf bifurcation is determined by

In this section, we present some numerical simulations to verify our theoretical results proved in previous sections by using MATLAB DDE solver. We simulate the system (

The phase portraits of system (

The trajectories and phase portrait of system (

The trajectories and phase portrait of system (

First, we choose parameters

Next, we choose parameters

In the present paper, we have considered a delayed predator-prey system in which the prey species exhibits herd behavior and the predator species with quadratic mortality. Our research shows that, for system (

Our results may enrich the dynamics in the predator-prey system and help us to better understand the interaction of predator with prey in a real ecosystem. Further studies are necessary to analyze the dynamics of more realistic but complex systems, such as delayed diffusive predator-prey system with herd behavior.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to thank the handling editor and reviewers for their valuable comments and suggestions which lead to truly significant improvement of the paper. This research is supported by the National Natural Science Foundation of China (no. 11271260), Shanghai Leading Academic Discipline Project (no. XTKX2012), Innovation Program of Shanghai Municipal Education Commission (no. 13ZZ116), and the Innovation Fund Project For Graduate Student of Shanghai (no. JWCXSL1401).