Ecological interactions between species that prefer different habitat types but come into contact in edge regions at the interfaces between habitat types are modeled via reaction-diffusion systems. The primary sort of interaction described by the models is competiton mediated by pathogen transmission. The models are somewhat novel because the spatial domains for the variables describing the population densities of the interacting species overlap but do not coincide. Conditions implying coexistence of the two species or the extinction of one species are derived. The conditions involve the principal eigenvalues of elliptic operators arising from linearizations of the model system around equlibria with only one species present. The conditions for persistence or extinction are made explicit in terms of the parameters of the system and the geometry of the underlying spatial domains via estimates of the principal eigenvalues. The implications of the models with respect to conservation and refuge design are discussed.

We consider delay differential equations describing the dynamics of isolated neurons. First, we review some existing results on the existence and stability of equilibria and periodic solutions, which are closely related to pattern formation. Then, one result on pattern formation in periodic environment is improved by using the theory of coincidence degree.

This research is in collaboration with Robert Stephen Cantrell and Yuan Lou. We consider a diffusive Lotka-Volterra competition model on a bounded domain and ask how varying the boundary conditions from Neumann through Robin to Dirichlet affects the dynamics of the model. We show that by changing the boundary conditions the predictions of the model can be changed from dominance by the first competitor and extinction of the second to coexistence, then to dominance by the second competitor and extinction of the first, then back to coexistence, then back to dominance by the first competitor, and this switching can occur several times. To construct examples of this phenomenon we start with a situation where the two species are identical and the coefficients are constant and then perturb the system with an appropriate spatially heterogeneous term in one of the growth rates. The mathematical methods underlying the analysis include bifurcation/continuation theory and various more or less classical ideas from the theory of differential equations.

Whenever the development of an individual is affected by a quantity such as food which, in turn, is affected by the individuals (consumption!), the resulting population model is quasilinear. If we cut the feedback loop, solve a parameterized non-autonomous linear problem and then restore the loop, we obtain a fixed point problem that can be solved under appropriate assumptions. This methodology yields a constructive definition of an infinite-dimensional dynamical system. The qualitative theory of such systems is still very much in its infancy and accordingly there are many open problems. The lecture is based on joint work (over a large number of years) with Mats Gyllenberg, Haiyang Huang, Markus Kirkilionis, Hans Metz and Horst Thieme.

We show that for certain population models, heterogeneous environments depending on a small parameter can be designed so that as the small parameter goes to zero, the distributions of the population exhibit clear prescribed spatial patterns in the environment. This will be demonstrated in the single species logistic model and the two species Lotka-Volterra competition model.

We study the asymptotic behavior of positive solutions of a semilinear parabolic equation with a nonlinear boundary condition. This problem admits a unique stationary solution which is not bounded and attracts all positive solutions. We find their growth rate at the singular point on the boundary. This is a joint work with J. J. L. Velazquez and M. Winkler.

Traditionally epidemic spread in space is modeled either by diffusion equations or by contact distributions. Evidently contacts between individuals at different positions require that individuals move. Here it is shown that contact models can be derived as limiting cases from diffusion models with two levels of spatial mixing by appropriate scaling of the parameters. This approach can be used in a much wider range of problems: Many populations, human and animal, show two rather distinct migration patterns, not only humans move around in their neighborhood and occasionally travel over long distances. In an appropriate scaling the neighborhood shrinks to a point, individuals switch between a sedentary and a migrating state. A key example showing the effects of the migration pattern together with the nonlinearity is Fisher's equation with a resting state, i.e., a scalar reaction diffusion equation coupled to an ordinary differential equation, formally similar to the Fitzhugh-Nagumo system but with very different properties.

In this talk I will propose a model for aggregating individuals through a random walk with bias toward scent marks which are made the individuals. The probability density function for an individual is given as the solution to a coupled system of partial/ordinary differential equations. This model differs from chemotaxis equations, where the scent marks can move. The analysis of the system, via application of an energy method, leads to distinct aggregations with abrupt edges. Applications will be made to the formation of territories, and connections will be made to ecological models for aggregating populations (e.g. Turchin 1989, Journal of Animal Ecology 58: 75-100).

The Langevin equation for individual movement and the corresponding PDE for the (probability) density, known as the Kramers equation, have been used widely in the realm of physics. In biological modeling, however, the location jump process and the corresponding diffusion equation are the most common tools. In this talk, we briefly discuss the underlying assumptions of both approaches to conclude that the Langevin approach might be more appropriate in many cases in biological modeling. We then use scalings to find conditions under which the simpler diffusion equation approximates the more complicated Kramers equation. We then prove an approximation theorem using the moments of the Kramers equation. Next, we introduce a birth-death process into Kramers equation to describe population growth and spread. We show that the resulting reaction Kramers equation (analogous to reaction diffusion equation) has a unique global solution under some conditions on the reaction term. Finally, we apply the modeling framework to chemotaxis. We show that a certain class of chemotaxis equation has the same form as the well known Vlasov-Poisson-Fokker-Planck system from statistical mechanics, and that the moment closure procedure introduced above leads to the classical Patlak-Keller-Segel model. This is joint work with K.P. Hadeler and T. Hillen.

We exhibit several interesting properties of solutions of semilinear parabolic equations on R^N. Even though the solutions we examine have compact trajectories and decay to zero at spatial infinity uniformly with respect to time, they behave much differently from solutions of the Dirichlet problem on bounded domains. Issues to be discussed include quasiconvergence (convergence to a set of equilibria) and examples without asymptotic symmetrization.

Long-wave unstable thin film equations

It is well accepted that the growth of a tumour is dependent on its ability to induce the growth of new blood vessels; a process called angiogenesis. Hopes have been raised that an anti-angiogenic treatment may be effective in the fight against cancer. We formulate, using the theory of reinforced random walks, an individual cell-based model of angiogenesis. The anti-angiogenic potential of angiostatin, a known inhibitor of angiogenesis, is also examined.The capillary networks predicted by the model are in good qualitative agreement with experimental observations.

Though Alan Turing predicted in 1952 that chemicals can react and diffuse in such a way to destabilize a homogeneous stationary state, and result in inhomogeneous spatial patterns, the first experimental evidence for Turing structures was only observed in 1990 in the chlorite-iodide-malonic acid-starch (CIMA) reaction. In this talk I will describe some fundamental properties of Turing patterns, through our mathematical analysis for the Lengyel-Epstein model, a two-variable reaction diffusion system which captures the crucial feature of the reaction.

In the present paper we consider the nonlinear evolution equation u¢+Au ' G(u), where A: D(A) Í X® X is m-accretive with (I+lA)^-1 compact for some l > 0, and G:[`(D(A))]® X is continuous, and we prove that the orbit {u(t); t Î R₊} is relatively compact if and only if u is uniformly continuous, and both u and G °u are bounded on R₊. In the same spirit, we derive conditions for orbits of bounded sets to have compact attractors. Some consequences and an example from age-structured population dynamics illustrate the effectiveness of the abstract result.

In this talk, I will report recent progress towards understanding the ground state solutions for the following Gierer-Meinhardt system in R^N, N ³ 2:

Almost automorphy is a notion first introduced by S. Bochner in 1955 to generalize the almost periodic one. It is proven to be a fundamental notion in characterizing multi-frequency phenomena and their generating dynamical complexity. This lecture will discuss the existence of almost automorphic spatial dynamics in lattice differential equations and their role played in the onset of the pattern formation and spatial chaos.

The saddle point behavior is established for monotone semiflows with weak bistability structure and then these results are applied to three reaction-diffusion systems modelling man-environment-man epidemics, single loop positive feedback and two species competition, respectively.

We consider periodic neutral functional differential equations. By combining the theory of monotone semiflows generated by neutral functional differentail equations with Krasnosel'skii's fixed point theorem, we establish sufficient conditions for existence, uniqueness and global attractivity of a periodic solution of the equation. We also apply the results to a concrete neutral equation that models single-species growth, the spread of epidemics, and the dynamics of capital stocks in peirodic environment.