Kohkichi Kawasaki

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.

MODELING STRATIFIED DIFFUSION IN BIOLOGICAL INVASIONS

Recent data on biological invasion show that range expansion is driven by various modes of dispersal such as neighborhood diffusion and long-distance dispersal that occur side by side within a species. In such a stratified dispersal process, the initial range expansion mainly occurs by neighborhood diffusion. However, as the range of the founder population expands, new colonies created by long-distance migrants increase in number to cause an accelerating range expansion in the later phase. We classify several well-documented examples of geographical expansions into three major types depending on the nonlinearity of the range-versus-time curve. To examine how long-distance dispersal produces accelerating range expansion, we construct a stratified diffusion model, which describes the dynamics of the size distribution of colonies created by long-distance migrants. The model consists of a von Foerster equation combined with a Skellam model. Analyzing the model provides an estimate of range expansion in terms of the rate of expansion due to neighborhood diffusion, the leap distance, and the colonization rate of long-distance migrants. The results explain various types of nonlinear range expansion observed in biological invasions.

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.

Spatial segregation in competitive interaction-diffusion equations

SummaryThe effect of cross-population pressure on the Volterra type dynamics for two competing species is investigated. On the basis of cross-diffusion induced instability, spatial segregation is studied. Spatially discrete models are also discussed. It is shown that this effect has a tendency to enhance the stability assuring coexistence of species. In continuous and discrete cases, time-dependent segregation processes are studied numerically.

Modeling the Population Dynamics of a Cuckoo-Host Association and the Evolution of Host Defenses

Cuckoo parasitism in Nagano Prefecture in Japan has shown dramatic changes in the parasitism rate, host usage by the cuckoo, and defensive behavior of hosts during the past 60 yr. To gain insights into these phenomena, we model the population dynamics of a cuckoo-host association together with the population genetics of a rejecter gene in the host population. Analysis shows that both the dynamical change in the host-parasite association and the establishment of the hosts counteradaptation crucially depend on the product of two factors, the carrying capacity of the host and cuckoos searching efficiency. When the product is less than a critical value, the host population cannot evolve a counteradaptation even if parasitized by the cuckoo. Hence, the lack of counteradaptation does not necessarily imply that the host population only recently has become parasitized. As the product becomes larger, the rejection behavior will be eventually established at higher levels in the host population In this case, the spreading of rejection behavior is very fast, which suggests that the cuckoo-host association reaches an equilibrium state within a relatively short period. These results make possible new interpretations of several circumstances reported about cuckoo-host associations.

Modeling biological invasions into periodically fragmented environments.

Range expansion of a single species in a regularly striped environment is studied by using an extended Fisher model, in which the rates of diffusion and reproduction periodically fluctuate between favorable and unfavorable habitats. The model is analyzed for two initial conditions: the initial population density is concentrated on a straight line or at the origin. For each case, we derive a mathematical formula which characterizes the spatio-temporal pattern of range expansion. When initial distribution starts from a straight line, it evolves to a traveling periodic wave (TPW), whose frontal speed is analytically determinable. When the range starts from the origin, it tends to expand radially at a constant average speed in each direction (ray speed) keeping its frontal envelope in a similar shape. By examining the relation between the ray speed and the TPW speed, we derive the ray speed in a parametric form, from which the envelope of the expanding range can be predicted. Thus we analyze how the pattern and speed of the range expansion are affected by the pattern and scale of fragmentation, and the qualities of favorable and unfavorable habitats. The major results include: (1). The envelope of the expanding range show a variety of patterns, nearly circular, oval-like, spindle-like, depending on parameter values; (2). All these patterns are elongated in the direction of stripes; (3). When the scale of fragmentation is enlarged without changing the relative spatial pattern, the ray speed in any direction increases, i.e., the rate of range expansion increases.

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.

MODELING THE SPREAD OF PINE WILT DISEASE CAUSED BY NEMATODES WITH PINE SAWYERS AS VECTOR

An epidemic of pine wilt disease has been spreading in wide areas of Japan for nearly a century. The disease is caused by the pinewood nematode, Bursaphelenchus xylophilus, with the pine sawyer, Monochamus alternates, as vector. The spread of disease is facilitated by an obligatory mutualism between the nematode and the pine sawyer: the pine sawyer helps the nematode transmit to a new host tree, while the nematode supplies the pine sawyer with newly killed trees on which to lay eggs. We present a mathematical model to describe the host-vector interaction between pines and pine sawyers carrying nematodes, on the basis of detailed data on the population dynamics of pine sawyers and the incidence of pine wilt disease at a study site located on the northwest coast of Japan. We used the model to simulate the dynamics of the disease and predict how the epidemic could be controlled by eradication of the pine sawyer. The main results are as follows: (1) There is a minimum pine density below which the disease always fails in invasion. However, even if the pine density exceeds this minimum, the disease fails in invasion due to the Allee effect when the density of pine sawyers is very low. (2) The minimum pine density increases disproportionately with increase in the eradication rate. (3) The probability that a healthy tree will escape from infection until the epidemic dies out decreases sharply with increase in the initial pine density or the initial density of pine sawyers.

Modeling the Expansion of an Introduced Tree Disease

Pine wilt disease is caused by the introduced pinewood nematode, Bursaphelenchusxylophilus, for which the vector is the pine sawyer beetle, Monochamus alternatus. Native Japanese pines, black pine (Pinus thunbergii) and red pine (P. densiflora), are extremely sensitive to the nematodes infection, and the parasite has been expanding nationwide in the last few decades, despite intensive control efforts. To understand the parasites range expansion in Japan, we modeled the dynamics of the pines and the beetle that disperses the nematode, using an integro-difference equation in a one-dimensional space. Based on field data collected in Japan, we investigated the dependence of the parasites rate of range expansion on the eradication rate of the beetle, the initial pine density, and the beetle dispersal ability. Our model predicts several results. (1) The Allee Effect operates on beetle reproduction, and consequently the parasite cannot invade a pine stand, once the beetle density decreases below a threshold. (2) The distribution of the dispersal distance of the beetles critically affects the expansion rate of the disease. As the fraction of the beetles that travel over long distance increases from zero, the range expansion accelerates sharply. (3) However, too frequent long-range dispersal results in a failure of the parasite invasion due to the Allee Effect, suggesting the importance of correctly assessing the beetles mobility to predict the speed of range expansion of the parasite. (4) As the eradication rate is increased, the range expansion speed decreases gradually at first and suddenly drops to zero at a specific value of the eradication rate.

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.