Some exact solutions of a generalized Fisher equation related to the problem of biological invasion.

The problem of biological invasion in a model single-species community is considered, the spatiotemporal dynamics of the system being described by a modified Fisher equation. For a special case, we obtain an exact solution describing self-similar growth of the initially inhabited domain. By comparison with numerical solutions, we show that this exact solution may be applicable to describe an early stage of a biological invasion preceding the propagation of the stationary travelling wave. Also, the exact solution is applied to the problem of critical aggregation to derive sufficient conditions of population extinction. Finally, we show that the solution we obtain is in agreement with some data from field observations.

A dynamical model for the growth and size distribution of multiple metastatic tumors.

Metastasis is the spread of tumors culminating in the establishment of one or more secondary tumors at remote sites. In deciding the best treatment for cancer therapy, estimations of the colony size of metastatic tumors and predictions of the future spread of colonies are needed. A dynamical model for the colony size distribution of multiple metastatic tumors is presented here. The dynamics is described by equations that incorporate both the colonization by metastasis and the growth of each colony. When the colony growth is subject to the Gompertz function, the explicit solution obtained tends to an asymptotic stable distribution that shows a monotonically decreasing or U-shaped pattern according to the values of clinically significant parameters, such as the colonization coefficient and the fractal dimension of blood vessels. This predicted colony size distribution agrees well with successive data of a clinically observed size distribution of multiple metastatic tumors of liver cancer. The combined analysis of the theoretical colony size distribution and clinical data will give useful information on the diagnosis and the therapy for cancer patients.

Population dynamics of nitrifying bacteria in an aquatic ecosystem.

In a space environment such as Space Shuttle or Space Station, animal experiments with aquatic species in a closed system pose a crucial problem in maintaining their water quality for a long term. In nature, ammonia as an animal wastes is converted by nitrifying bacteria to nitrite or nitrate compounds, which usually become nitrogen sources for plants. Thus an application of the biological reactor with such bacteria attached on some filters has been suggested and experimentally studied for efficient waste managements of ammonia. Although some successful results were reported (Kozu et al. 1995, Nagaoka et al. 1998, Nakamura et al. 1997, 1998) in the space applications, purely empirical approaches have so far been taken to develop a biological filter having a stable nitrifying activity. In this study, we constructed a mathematical model to deal with the dynamics of the ammonia nitrifying processes in a biological reactor. The model describes population dynamics of the ammonia-oxidizing bacteria and the nitrite-oxidizing bacteria cultivated on the same filter. We estimated parameters involved in the model using the experimental data. The result shows that these estimated parameters could be applied to general cases and that the two bacteria are in a symbiotic relationship; they can better perform when both coexist, as has been empirically recognized. Based on the model analysis, we discuss how to prepare a high performance biological filter.

Modeling spatio-temporal patterns generated by Bacillus subtilis.

Colonies of bacteria, Bacillus subtilis, that grow on the surface of thin agar plates show various morphological patterns in response to environmental conditions, such as the nutrient concentration, the solidity of an agar medium and temperature. For instance, the colony pattern shows a dense-branching morphology with a smooth circular envelope (DBM-like) in a nutrient-poor semi-solid agar medium, and it turns to a simple disk-like colony as both the nutrient concentration and the agar's softness increase. These patterns have been shown to involve cell movement inside colonies. In a DBM-like colony, individual cells actively move, particularly in the expanding periphery of the colony, while they become immotile at the inner region of the colony where nutrient is very low. In a disk-like colony, cells are highly active in the whole region of the colony. Based on such experimental observations, we develop a diffusion-reaction model, in which density dependent cell movements are incorporated by the level of nutrient concentration available for the cell. Numerical simulations of the model under different environmental conditions closely reproduce various colony patterns ranging from DBM-like pattern to the homogeneous disk-like one in a unifying manner. The analysis also predicts the growth velocity of a colony as a function of the nutrient concentration.

Travelling Wave Solutions for the Spread of Farmers into a Region Occupied by Hunter-Gatherers

The geographical spread of an initially localized population of farmers into a region occupied by hunter-gatherers is modelled as a reaction-diffusion process in an infinite linear habitat. Hunter-gatherers who come in contact with farmers are converted at rate e. Growth of initial farmers, converted farmers, and hunter-gatherers is logistic with intrinsic rates rf, rc, and r h. The carrying capacities of all farmers (i.e., initial and converted farmers combined) and hunter-gatherers are K and L. Individuals migrate at random, where the diffusion constant, D, is the same for all three groups. Under the above assumptions, we deduce the conditions under which wavefronts of initial or converted farmers are generated. Numerical work suggests that a travelling wave solution of constant shape always exists, comprising an advancing wavefront of all farmers and a retreating wavefront of hunter-gatherers. Linear analysis in phase space suggests that the speed is given by 2(Drf)1/2 or 2[D(r c+eL)]1/2, whichever is greater. The composition of the expanding distribution of all farmers appears to depend on the relative magnitude of rf versus rc+eL and of eK versus rh. A wavefront of initial farmers is not generated if rfrh. On the other hand, a wavefront of initial farmers is generated if rf>rc+eL. In this case, the waveform is peaked with leading and trailing edges that converge to 0 if eKrh so that there is substantial displacement of the indigenous hunter-gatherers by the initial farmers.

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 host's counteradaptation crucially depend on the product of two factors, the carrying capacity of the host and cuckoo's 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.

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

An 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.

Analyzing insect movement as a correlated random walk.

This paper develops a procedure for quantifying movement sequences in terms of move length and turning angle probability distributions. By assuming that movement is a correlated random walk, we derive a formula that relates expected square displacements to the number of consecutive moves. We show this displacement formula can be used to highlight the consequences of different searching behaviors (i.e. different probability distributions of turning angles or move lengths). Observations of Pieris rapae (cabbage white butterfly) flight and Battus philenor (pipe-vine swallowtail) crawling are analyzed as a correlated random walk. The formula that we derive aptly predicts that net displacements of ovipositing cabbage white butterflies. In other circumstances, however, net displacements are not well-described by our correlated random walk formula; in these examples movement must represent a more complicated process than a simple correlated random walk. We suggest that progress might be made by analyzing these more complicated cases in terms of higher order markov processes.

Spatial distribution of dispersing animals.

A 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.