Nanako Shigesada

Spatial Segregation of Interacting Species

Abstract The distributional pattern formation of the populations of two competitive species in heterogeneous environments is analyzed. In the mathematical formulation, a non-linear dispersive force due to mutual interferences of individuals and an environmental potential function are introduced as a behavioral version of Morisitas phenomenological theory of “Environmental density”. Mathematical analyses of effects of these forces give the result that the heterogeneity of the environment and the non-linear dispersive movements raise a spatial segregation of the populations of two similar and competing species and there is a possibility that this spatial segregation acts to stabilize the coexistence of two similar species, relaxing the interspecific competition.

Traveling periodic waves in heterogeneous environments

Abstract A model for a single species population which propagates in a heterogeneous environment in a one dimensional space is presented. The environment is composed of two kinds of patches with different diffusivities and intrinsic growth rates, which are alternately arranged along the spatial axis. From the stability analysis of the model, the invasion condition for a new migrating species is obtained in terms of the sizes of patches, diffusivities, and growth rates. When the parameters satisfy the invasion condition, the distribution of the population initially localized in a bounded area always evolves into a traveling periodic wave (TPW) with a constant speed. When the invasion condition is not satisfied, the population fails in invasion tending to extinction. The velocity of TPW is calculated by using a dispersion relation, and the effects of the heterogeneity of the environment on the velocity are discussed.

Switching effect of predation on competitive prey species

Abstract The fact that the predation pressure has a stabilizing effect on the community of competitive species is demonstrated by a mathematical model of two-preys and one-predator system which has the switching property of predation. By analyzing a dynamical system for these three species populations, it is shown that, in a wide range of parameter space, the system has stable coexisting equilibrium states and the manifold of stable stationary points exhibits a cusp catastrophe and there exist two stable stationary points in the cusp region in the parameter space. Thus, it has been shown that Causes competitive exclusion is actually relaxed by the switching mechanism of predation.

Spatial distribution of dispersing animals

SummaryA mathematical model for the dispersal of an animal population is presented for a system in which animals are initially released in the central region of a uniform field and migrate randomly, exerting mutually repulsive influences (population pressure) until they eventually become sedentary. The effect of the population pressure, which acts to enhance the dispersal of animals as their density becomes high, is modeled in terms of a nonlinear-diffusion equation. From this model, the density distribution of animals is obtained as a function of time and the initial number of released animals. The analysis of this function shows that the population ultimately reaches a nonzero stationary distribution which is confined to a finite region if both the sedentary effect and the population pressure are present. Our results are in good agreement with the experimental data on ant lions reported by Morisita, and we can also interpret some general features known for the spatial distribution of dispersing insects.

Switching effect on the stability of the prey-predator system with three trophic levels

The effect of predator switching on the stability of two- and three-trophic level systems is analyzed. It is assumed that the predator has a switching property characterized by two parameters: a measure of bias in switching response and the intensity of switching. In the two-trophic level system which consists of two prey and one predator species, the following two qualitative conclusions are derived: (1) the stabilizing influence of switching requires heterogeneity in prey intrinsic growth rates, and (2) the stabilizing effect is enhanced if the bias in switching response is toward the prey with a lower intrinsic growth rate. In the three trophic level system in which the consumption of one prey by the other prey is allowed, the analysis leads to three conclusions: (1) Too much exploitation of one prey by the other precludes the coexistence of the three species. (2) If a coexisting steady state exists and the death rate of the middle species is not very large, the system has a locally stable equilibrium point or a stable limit cycle so that all three species persist together. (3) When the top predator prefers the lower prey species to the middle species, then switching enhances the stability of the coexisting steady state as the consumption rate of the lowest level by the middle species increases. Finally it is shown for both systems mentioned above that the evolutionarily stable state with respect to switching parameters is dynamically stable.

The role of rapid dispersal in the population dynamics of competition

A dynamical equation for the spatial distribution of competing species that contains a growth term and a dispersal term is analyzed under the condition that the dispersal rate is sufficiently rapid compared to the growth rate. With this assumption, the equation becomes analogous to the niche-partitioning theory of MacArthur and Levins (1967, Amer. Natur. 101, 377–385) and thus provides a link between the theory of local niche partitioning and the theory of regional habitat segregation. The formulation is applied to a two species system consisting of a specialist and a generalist competing with each other in an environment composed of two different habitats. The analysis shows that dispersal due to directed movement toward a favorable habitat and density dependent random movement both facilitate coexistence.

The effects of interference competition on stability, structure and invasion of a multi-species system

SummaryAn interference competition model for a many species system is presented, based on Lotka-Volterra equations in which some restrictions are imposed on the parameters. The competition coefficients of the Lotka-Volterra equations are assumed to be expressed as products of two factors: the intrinsic interference to other individuals and the defensive ability against such interference. All the equilibrium points of the model are obtained explicitly in terms of its parameters, and these equilibria are classified according to the concept of sector stability. Thus survival or extinction of species at a stable equilibrium point can be determined analytically.The result of the analysis is extended to the successional processes of a community. A criterion for invasion of a new species is obtained and it is also shown that there are some characteristic quantities which show directional changes as succession proceeds.

Theory of Bimolecular Reaction Processes in Liquids

A theoretical approach to the problem of diffusion controlled bimolecular reactions 1s presented. In order to take into account the ti:ne correlation of reaction process of our many-particle system, the probability of the fust reaction is introduced as a fundamental quantity. Time development of the ensemble of our system is formulated using the probability of the first reaction. An approximation which reduces the general formula to a problem of Markov process is adopted. Then it is shown that, if we assume stationary reaction rate, the usual phenomenological kinetic equation, i.e. the so called law of mass action can be derived as the first order approximation, and as the second order approximation the deviation from the law of mass action is examined. For the general case, in order to obtain the probability of the first reaction in an explicit form, it becomes necessary to solve. the multidimensional diffusion equation with pair absorbing interactions, which is calculated using the binary collision expansion method.

Spatial Distribution of Rapidly Dispersing Animals in Heterogeneous Environments

Ecological models incorporating spatial heterogeneity of habitats are of profound importance in understanding the movements of organisms and their effects on the stability of spatial distributions of populations under natural circumstances. Equations describing the time development of the spatial distribution of a population in a heterogeneous environment fundamentally involve two terms, dispersal and growth, which are both functions of space. There have been several distinct approaches to the analysis of such models depending on the system under investigation and the type of method being applied (See reviews by Okubo (1980) and Levin (1981)). Among them, models for a single species in one dimensional space have been extensively studied for various types of ecological systems. Okubo (1980) analyzed effects of various kinds of spatially varying dispersal on the spatial structure of populations. Gurney and Nisbet (1974) and Namba (1980) included a spatially varying growth term in their model. In population genetics, Fleming (1975), Nagylaki (1975) and May et al. (1975) studied the effect of environmental heterogeneity on the viability of individuals of a single species and presented the condition for the existence of clines in a one-dimensional space. As for twospecies systems, the effect of dispersal with directed movements was taken into consideration by Comins and Blatt (1974) and Shigesada et al. (1979), and the effect of spatially varying growth was considered by Pacala and Roughgarden (1982) and Kawasaki and Teramoto (1979). However, these models incorporated the effect of heterogeneity either in dispersal or growth, but not in both processes.

The speeds of traveling frontal waves in heterogeneous environments

Since Fisher’s pioneering work (Fisher 1937), many studies on traveling waves in a growing population have been performed. The model proposed by Fisher consists of a diffusion equation with a logistic growth term: