Bifurcation Analysis Of Predator-Prey Models With Nonmonotonic Functional Response

Mathematical Biology is a fast-developing interdisciplinary subject.In this field, on one hand, researchers apply Mathematical methods to Biological research, unravel the underlying mechanism and solve problems arising from Biology, on the other hand, Biology opens up new Mathematics realms and stimulate the development of Mathematics theories. This relationship between Mathematics and Biology is vividly described by the headline of [4]: "Mathematics is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better."Population dynamics is one important branch of Ecology, and its development is systematic and relatively mature, involved with intensive Mathematical theory. Population dynamics studies the interactionbetween population and environment and between the differentpopulations and short- and long-term changes in size and other aspects. Biologists and ecologists set up different models accordingto different interaction, such as competition, predator-prey, mutualism or symbiosis and so on. These models are applied to description,prediction and control of the populations development. For more fundamentals on the subject, refer to [25, 38, 41].In population dynamics, the research on Predator-prey modelsis a main direction. In these models, a functional response of the predator to the prey density refers to the change in the densityof prey attached per unit time per predator as the prey density changes. In general, the response function is assumed as an continuouslydifferentiable function. However, this kind of response functionsis not precise in some cases, such as the inhibition in microbial dynamics, group defense and plant toxin in population dynamics. In these cases, nonmonotonic response functions are necessary to model the according phenomena.This paper is an overview on the development of Predator-PreyModels with nonmonotonic functional response. It mainly discusses the models nonmonotonic functional response caused by the group defense and plant toxin, and analyzes the possible bifurcations when the parameters of the models change. In chapter 2, preliminary conceptsand theorems on bifurcations and predator-prey models are introduced.Chapter 3 mainly discusses three predator-prey models with nonmonotonc response function due to group defense[1]. In section 3.1, a generalized Gause type model[13][24][35]is set up, where x(t), y(t) denote the density of prey and predatorpopulations, g, p are continuously differentiable functions and K, d, c denote the carrying capacity of prey, the death rate of predatorand the conversion rate, which are positive constants. g(x, K) is the specific growth rate of the prey in the absence of predation, which is assumed to have some properties, and g(x, K) = r(1—x/K) is a prototype. p(x) is the response function, which is a concave function with one hump. p(x) = mx/(ax~2 + bx + 1) and p(x) = mx/(a+x~2) satisfy the assumptions on p(x). Then, [35] analyzes the stability of all possible equilibria and the possible bifurcation when K varies, and indicates the existence of Hopf bifurcation and homo-clinic bifurcation at some K by means of Hopf bifurcation theorem and portrait analysis. However, [35] does not analyze the possible degenerate equilibrium when the death rate of predator is also chosenas bifurcation parameter. Later, Ruan and Xiao[29] consider the model with simplified Holling TypeⅣresponse function, and the intrinsicgrowth rate of prey is Logistic function g(x, K) = r(1-x/K). The model iswhich is analyzed in section 3.2. Apart from the linear analysis of equilibria, section 3.2 shows that there exists supercritical, subcriticaland degenerate Hopf bifurcation surfaces depending all parametersof the model. In addition, [29] shows the cusp point, which is the unique interior equilibrium when parameters take some values,is the Bogdanov-Takens bifurcation point of codimension 2. Then, it shows the model is the generic unfolding of the model at the cusp point and exhibits Bogadnov-Takens bifurcation by transformingthe model into normal form. In section 3.3, we introduce the work[37], which takes the Holling TypeⅣfunction as the response function and considers the following model[37] indicates there exists a Bogdanov-Takens bifurcation of codimension3 by taking K, d, b as bifurcation parameters, which does not happens in the model with simplified Holling TypeⅣresponse function. When b＞- 2(?) and b≠-(?), it has a Bogdanov-Takens bifurcation of codimension 2 in the neighborhood of the cusp. Particularly,when b = 0, the model is analyzed by [29]. Thus, [37] extends the results of [29].In chapter 4, plant toxin is taken into consideration by modifyingHolling type II response function which is commonly employed to describe the plant consumption by the herbivore. Plant toxin has an negative effect on the herbivore and can lead to a decrease in the growth rate when the plant density is high[10] [11] [20] [21] [32]. This research is the latest development of the nonmonotonic functional response functions. The 2-dimensional plant-herbivore model iswhere N, P denote the density of plant and the density of herbivore,d,K are positive constants and C(N) = f(N)(1 - f(N)/4G) is the revised HollingⅡfunctional response function, in which G measures the toxicity level. C(N) is a function with one hump, i.e. a nonmonotonic function in this chapter. G, d are chosen as bifurcationparameters. After the analysis of the stability of equilibria, one Hopf bifurcation curve is determined. The model has a cusp point of codimension 2, which is proven to be a Bogdanov-Takens bifurcationpoint. The Bogdanov-Takens bifurcation indeed happens at some d, K by using the similar method in 3.2 and 3.3.