Nonlinear Diffusion for Bacterial Traveling Wave Phenomenon.

The bacterial traveling waves observed in experiments are of pulse type which is different from the monotone traveling waves of the Fisher-KPP equation. For this reason, the Keller-Segel equations are widely used for bacterial waves. Note that the Keller-Segel equations do not contain the population dynamics of bacteria, but the population of bacteria multiplies and plays a crucial role in wave propagation. In this paper, we consider the singular limits of a linear system with active and inactive cells together with bacterial population dynamics. Eventually, we see that if there are no chemotactic dynamics in the system, we only obtain a monotone traveling wave. This is evidence that chemotaxis dynamics are needed even if population growth is included in the system.

An aggregation model of cockroaches with fast-or-slow motion dichotomy.

We propose a mathematical model, namely a reaction-diffusion system, to describe social behaviour of cockroaches. An essential new aspect in our model is that the dispersion behaviour due to overcrowding effect is taken into account as a counterpart to commonly studied aggregation. This consideration leads to an intriguing new phenomenon which has not been observed in the literature. Namely, due to the competition between aggregation towards areas of higher concentration of pheromone and dispersion avoiding overcrowded areas, the cockroaches aggregate more at the transition area of pheromone. Moreover, we also consider the fast reaction limit where the switching rate between active and inactive subpopulations tends to infinity. By utilising improved duality and energy methods, together with the regularisation of heat operator, we prove that the weak solution of the reaction-diffusion system converges to that of a reaction-cross-diffusion system.

Traveling waves for a three-component reaction-diffusion model of farmers and hunter-gatherers in the Neolithic transition.

The Neolithic transition began the spread of early agriculture throughout Europe through interactions between farmers and hunter-gatherers about 10,000 years ago. Archeological evidences indicate that the expanding velocity of farming into a region occupied by hunter-gatherers is roughly constant all over Europe. In the late twentieth century, from the contribution of the radiocarbon dating, it could be found that there are two types of farmers: one is the original farmer and the other is the converted farmer which is genetically hunter-gatherers but learned agriculture from neighbouring farmers. Then this raises the following questions: Which farming populations play a key role in the expansion of farmer populations in Europe? and what is the fate of hunter-gatherers (e.g., become extinct, or live in lower density, or live in agricultural life-style)? We consider a three-component reaction-diffusion system proposed by Aoki, Shida and Shigesada, which describes the interactions among the original farmers, the converted farmers, and the hunter-gatherers. In order to resolve these two questions, we discuss traveling wave solutions which give the information of the expanding velocity of farmer populations. The main result is that two types of traveling wave solutions exist, depending on the growth rate of the original farmer population and the conversion rate of the hunter-gatherer population to the converted farmer population. The profiles of traveling wave solutions indicate that the expansion of farmer populations is determined by the growth rate of the original farmer and the (maximal) carrying capacity of the converted farmer, and the fate of hunter-gatherers is determined by the growth rate of the hunter-gatherer and the conversion rate of the hunter-gatherer to the converted farmer. Thus, our results provide a partial answer to the above two questions.

Complex pattern formation driven by the interaction of stable fronts in a competition-diffusion system.

The ecological invasion problem in which a weaker exotic species invades an ecosystem inhabited by two strongly competing native species is modelled by a three-species competition-diffusion system. It is known that for a certain range of parameter values competitor-mediated coexistence occurs and complex spatio-temporal patterns are observed in two spatial dimensions. In this paper we uncover the mechanism which generates such patterns. Under some assumptions on the parameters the three-species competition-diffusion system admits two planarly stable travelling waves. Their interaction in one spatial dimension may result in either reflection or merging into a single homoclinic wave, depending on the strength of the invading species. This transition can be understood by studying the bifurcation structure of the homoclinic wave. In particular, a time-periodic homoclinic wave (breathing wave) is born from a Hopf bifurcation and its unstable branch acts as a separator between the reflection and merging regimes. The same transition occurs in two spatial dimensions: the stable regular spiral associated to the homoclinic wave destabilizes, giving rise first to an oscillating breathing spiral and then breaking up producing a dynamic pattern characterized by many spiral cores. We find that these complex patterns are generated by the interaction of two planarly stable travelling waves, in contrast with many other well known cases of pattern formation where planar instability plays a central role.

Ecological invasion in competition-diffusion systems when the exotic species is either very strong or very weak.

Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among m ecological species for a limited resource necessary to their survival and growth. Notwithstanding their rather simple mathematical structure, such systems may display quite interesting behaviours. In particular, while for [Formula: see text] no coexistence of the two species is usually possible, if [Formula: see text] we may observe coexistence of all or a subset of the species, sensitively depending on the parameter values. Such coexistence can take the form of very complex spatio-temporal patterns and oscillations. Unfortunately, at the moment there are no known tools for a complete analytical study of such systems for [Formula: see text]. This means that establishing general criteria for the occurrence of coexistence appears to be very hard. In this paper we will instead give some criteria for the non-coexistence of species, motivated by the ecological problem of the invasion of an ecosystem by an exotic species. We will show that when the environment is very favourable to the invading species the invasion will always be successful and the native species will be driven to extinction. On the other hand, if the environment is not favourable enough, the invasion will always fail.

On a nonlocal system for vegetation in drylands.

Several mathematical models are proposed to understand spatial patchy vegetation patterns arising in drylands. In this paper, we consider the system with nonlocal dispersal of plants (through a redistribution kernel for seeds) proposed by Pueyo et al. (Oikos 117:1522-1532, 2008) as a model for vegetation in water-limited ecosystems. It consists in two reaction diffusion equations for surface water and soil water, combined with an integro-differential equation for plants. For this system, under suitable assumptions, we prove well-posedness using the Schauder fixed point theorem. In addition, we consider the stationary problem from the viewpoint of vegetated pattern formation, and show a transition of vegetation patterns when parameter values (rainfall, seed dispersal range, seed germination rate) in the system vary. The influence of the shape of the redistribution kernel is also discussed.

Parameter Scaling for Epidemic Size in a Spatial Epidemic Model with Mobile Individuals.

In recent years, serious infectious diseases tend to transcend national borders and widely spread in a global scale. The incidence and prevalence of epidemics are highly influenced not only by pathogen-dependent disease characteristics such as the force of infection, the latent period, and the infectious period, but also by human mobility and contact patterns. However, the effect of heterogeneous mobility of individuals on epidemic outcomes is not fully understood. Here, we aim to elucidate how spatial mobility of individuals contributes to the final epidemic size in a spatial susceptible-exposed-infectious-recovered (SEIR) model with mobile individuals in a square lattice. After illustrating the interplay between the mobility parameters and the other parameters on the spatial epidemic spreading, we propose an index as a function of system parameters, which largely governs the final epidemic size. The main contribution of this study is to show that the proposed index is useful for estimating how parameter scaling affects the final epidemic size. To demonstrate the effectiveness of the proposed index, we show that there is a positive correlation between the proposed index computed with the real data of human airline travels and the actual number of positive incident cases of influenza B in the entire world, implying that the growing incidence of influenza B is attributed to increased human mobility.

Evolutionary branching and evolutionarily stable coexistence of predator species: Critical function analysis.

On the ecological timescale, two predator species with linear functional responses can stably coexist on two competing prey species. In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate under what conditions such a coexistence is also evolutionarily stable, and whether the two predator species may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in capture rates and a predator with a high capture rate for one prey has a low capture rate for the other and vice versa. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly convex in the vicinity of the singular strategy and the interspecific prey competition is not strong, then this singular strategy is an evolutionary branching point, near which the resident and mutant predator populations can coexist and diverge in their strategies. Second, we find that after branching has occurred in the predator phenotype, if the trade-off curve is globally convex, the predator population will eventually branch into two extreme specialists, each completely specializing on a particular prey species. However, in the case of smoothed step function-like trade-off, an interior dimorphic singular coalition becomes possible, the predator population will eventually evolve into two generalist species, each feeding on both of the two prey species. The algebraical analysis reveals that an evolutionarily stable dimorphism will always be attractive and that no further branching is possible under this model.

Adaptive evolution of foraging-related traits in a predator-prey community.

In this paper, with the method of adaptive dynamics and geometric technique, we investigate the adaptive evolution of foraging-related phenotypic traits in a predator-prey community with trade-off structure. Specialization on one prey type is assumed to go at the expense of specialization on another. First, we identify the ecological and evolutionary conditions that allow for evolutionary branching in predator phenotype. Generally, if there is a small switching cost near the singular strategy, then this singular strategy is an evolutionary branching point, in which predator population will change from monomorphism to dimorphism. Second, we find that if the trade-off curve is globally convex, predator population eventually branches into two extreme specialists, each completely specializing on a particular prey species. However, if the trade-off curve is concave-convex-concave, after branching in predator phenotype, the two predator species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist. The analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible under this model.

The evolution of phenotypic traits in a predator-prey system subject to Allee effect.

This paper considers the evolution of phenotypic traits in a community comprising the populations of predators and prey subject to Allee effect. The evolutionary model is constructed from a deterministic approximation of the stochastic process of mutation and selection. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that the strong Allee effect of prey facilitates the formation of continuously stable strategy in the case that prey population undergoes evolutionary branching if the Allee effect of prey is not strong enough. Secondly, we show that evolutionary suicide is impossible for prey population when the intraspecific competition of prey is symmetric about the origin. However, evolutionary suicide can occur deterministically on prey population if prey individuals undergo strong asymmetric competition and are subject to Allee effect. Thirdly, we show that the evolutionary model with symmetric interactions admits a stable limit cycle if the Allee effect of prey is weak. Evolutionary cycle is a likely outcome of the process, which depends on the strength of Allee effect and the mutation rates of predators and prey.

Synergistic effect of two inhibitors on one activator in a reaction-diffusion system.

We investigate a three-component reaction-diffusion system that describes the interaction of one activator and two inhibitors where one inhibitor acts as a traveling pulse generator of the activator and the other acts as a lateral inhibition localizer. It is numerically shown that the synergistic effect of these two inhibitors on one activator induces several spatiotemporal patterns such as destabilization and nonannihilation of traveling pulses and the occurrence and splitting of traveling spots. By using singular perturbation procedures, the stability of radially symmetric equilibrium solutions is discussed. Furthermore, we discuss how such dynamics are caused under the synergistic effect of two inhibitors.

Diffusion, cross-diffusion and competitive interaction.

The cross-diffusion competition systems were introduced by Shigesada et al. [J. Theor. Biol. 79, 83-99 (1979)] to describe the population pressure by other species. In this paper, introducing the densities of the active individuals and the less active ones, we show that the cross-diffusion competition system can be approximated by the reaction-diffusion system which only includes the linear diffusion. The linearized stability around the constant equilibrium solution is also studied, which implies that the cross-diffusion induced instability can be regarded as Turing's instability of the corresponding reaction-diffusion system.

Nonannihilation dynamics in an exothermic reaction-diffusion system with mono-stable excitability.

We consider a 2-component excitable and diffusive system which describes a simple exothermic reaction process. In some parameter regime, there are two characteristics of travelling pulses of the system: (i) travelling pulses are planarly unstable; (ii) when two travelling pulses approach closely, they do not annihilate each other and repel like elastic objects. Under this situation, it is shown that ring patterns break down into complex patterns in 2-dimensions, which are totally different from those arising in the well-known excitable and diffusive system with the FitzHugh-Nagumo nonlinearity. (c) 1997 American Institute of Physics.