A MICROFOUNDATION OF PREDATOR‐PREY DYNAMICS

ABSTRACT. Predator‐prey relationships account for an important part of all interactions betweenspecies. In this paper we provide a microfoundation for such predator‐prey relations in afood chain. Basic entities of our analysis are representative organisms of species modeled similar to economic households. With prices as indicators of scarcity, organisms are assumed to behave as if they maximize their net biomass subject to constraints which express the organisms' risk of being preyed upon during predation. Like consumers, organisms face a ‘budget constraint’ requiring their expenditure on prey biomass not to exceed their revenue from supplying own biomass. Short‐run ecosystem equilibria are defined and derived. The net biomass acquired by the representative organism in the short term determines the positive or negative population growth. Moving short‐run equilibria constitute the dynamics of the predator‐prey relations that are characterized in numerical analysis. The population dynamics derived here turn out to differ significantly from those assumed in the standard Lotka‐Volterra model.     

扰动对天敌有多食性模型稳定性的影响

本文主要讨论了扰动对天敌具有多食性模型 x_1=x_1(r_1-a_1y), x_2=x_2(r_2-a_2y), y=y(-r_3+b_1x_1+b_2x_2)稳定性的影响。利用Liapunov函数得到了昆虫种类内部的密度制约,将促使昆虫与天敌系统进一步稳定化,进而得出同一食饵水平上的竞争是不稳定化的结论。其中后一种情况与J.M,Smith的结论完全一致。     

一类具有时滞捕食-食饵系统的稳定性分析

研究了一类带有时滞和对食饵具有阶段结构的捕食-食饵系统，通过线性化得到了该系统平衡点的局部稳定的充分条件．给出了解的渐近性质和阶段结构对种群持续生存的负面影响、     

基于比率的离散型捕食系统的周期解

针对一类基于比率的离散型捕食系统，利用重合度理论给出了该系统正周期解的存在性判据．     

基于改进的Monte Carlo方法及其应用

建立了一个基于网格由狼羊草组成的生态系统模型，模型中对捕食者(狼)及被捕食者(羊)给予了相应的属性定义，把草界定为可再生资源，通过引入“超结构”的规则，解决了采用Monte Carlo。方法模拟时网格状态不能同步更新的问题。通过模拟，得到模型的三种典型结果：即捕食者灭绝，捕食者被捕食者都灭绝以及捕食者被捕食者共存，给出了三种结果出现的几率分布，而且只要捕食者的初始密度不太大，生态系统很容易达到共存状态，生态系统对应的网格规模越大，生态系统越容易达到共存。     

Impact of proliferation strategies on food web viability in a model with closed nutrient cycle

A food web model with a closed nutrient cycle is presented and analyzed via Monte Carlo simulations. The model consists of three trophic levels, each of which is populated by animals of one distinct species. While the species at the intermediate level feeds on the basal species, and is eaten by the predators living at the highest level, the basal species itself uses the detritus of animals from higher levels as the food resource. The individual organisms remain localized, but the species can invade new lattice areas via proliferation. The impact of different proliferation strategies on the viability of the system is investigated. From the phase diagrams generated in the simulations it follows that in general a strategy with the intermediate level species searching for food is the best for the survival of the system. The results indicate that both the intermediate and top level species play a critical role in maintaining the structure of the system.     

Coexistence and harvesting control policy in a food chain model with mutual defense of prey

A model is proposed to understand the dynamics in a food chain (one predator‐two prey). Unlike many approaches, we consider mutualism (for defense against predators) between the two groups of prey. We investigate the conditions for coexistence and exclusion. Unlike Elettreby's (2009) results, we show that prey can coexist in the absence of predators (as expected since there is no competition between prey). We also show the existence of Hopf bifurcation and limit cycle in the model, and numerically present bifurcation diagrams in terms of mutualism and harvesting. When the harvest is practiced for profit making, we provide the threshold effort value that determines the profitability of the harvest. We show that there is zero profit when the constant effort is applied. Below (resp. above) , there will always be gain (resp. loss). In the case of gain, we provide the optimal effort and optimal steady states that produce maximum profit and ensure coexistence. Recommendations for resource managers As a result of our investigation, we bring the following to the attention of management:

• 1. In the absence of predators, different groups of prey can coexist if they mutually help each other (no competition among them).
• 2. There is a maximal effort to invest in order to gain profit from the harvest. Above , the investment will result in a loss.
• 3. In the case of profit from harvest, policy makers should recommend the optimal effort to be applied and the optimal stock to harvest. This will guarantee maximum profit while ensuring sustainability of all species.

     

Periodic solutions to a class of biological diffusion models with hysteresis effect

This paper is concerned with a class of biological models which consists of a nonlinear diffusion equation and a hysteresis operator describing the relationship between some variables of the equations. By the viscosity approach, we show the existence of periodic solutions of the problem under consideration. More precisely, with the help of the subdifferential operator theory and Leray–Schauder theorem, we show the existence of periodic solutions to the approximation problem and then obtain the solution of the original problem by using a passage-to-limit procedure.     

Delayed stability switches in singularly perturbed predator–prey models

In this paper we provide an elementary proof of the existence of canard solutions for a class of singularly perturbed planar systems in which there occurs a transcritical bifurcation of the quasi steady states. The proof uses the one-dimensional result proved by V.F. Butuzov, N.N. Nefedov and K.R. Schneider, and an appropriate monotonicity assumption on the vector field. The result is applied to identify all possible predator–prey models with quadratic vector fields allowing for the existence of canard solutions.