Effects of Behavioral Tactics of Predators on Dynamics of a Predator-Prey System

Table 1

The evolutionary stable state in LVM and AM.

𝑢𝐶<𝛼2𝑢+𝛼𝐶<𝑢3𝛼𝑎𝛼𝑎	LVM	AM	𝑢𝐶>𝛼𝑢<𝛼𝑎2𝑢+𝛼𝐶3𝛼𝑎	LVM	AM
𝐾<2𝑢+𝛼𝐶3𝛼𝑎	(𝐾,0)	(𝐾,0)	𝑢𝐾<𝛼𝑎	(𝐾,0)	(𝐾,0)
2𝑢+𝛼𝐶𝑢3𝛼𝑎<𝐾<𝛼𝑎	(𝐾,0)	(𝑛,𝑝)	𝑢𝛼𝑎<𝐾<2𝑢+𝛼𝐶3𝛼𝑎	(𝑛,𝑝)	(𝐾,0)
𝑢𝐾>𝛼𝑎	(𝑛,𝑝)	(𝑛,𝑝)	𝐾>2𝑢+𝛼𝐶3𝛼𝑎	(𝑛,𝑝)	(𝑛,𝑝)
	𝑛>𝑛	𝑝<𝑝		𝑛<𝑛	𝑝>𝑝

A comparison between evolutionary stable state of the models LVM and AM is given. Parameters 𝐶,𝐾,𝑢,𝛼,𝑎 represent fighting cost, carrying capacity of prey, natural mortality rate of prey, a conversion positive coefficient into biomass of predators, a positive parameter in functional response, respectively. Two cases are recognized by considering whether 𝐶 is smaller than 𝑢/(𝛼𝑎) or not. Furthermore, parameter region {𝐾,𝑢,𝛼,𝑎,𝐶} can be divided into three parts according to the evolutionary stable state in two models for each case. It shows that the stable state for two modes appears as either kept a constant with only prey at carrying capacity, or coexistence of prey and predators with a variation of population density, or a change between only prey at carrying capacity and coexistence of prey and predators. The values of (𝑛,𝑝) and (𝑛,𝑝) are given in the main text.