K. P. Hadeler

Predator-prey populations with parasitic infection.

A predator-prey model, where both species are subjected to parasitism, is developed and analyzed. For the case where there is coexistence of the predator with the uninfected prey, an epidemic threshold theorem is proved. It is shown that in the case where the uninfected predator cannot survive only on uninfected prey, the parasitization could lead to persistence of the predator provided a certain threshold of transmission is surpassed.

Epidemiological models for sexually transmitted diseases

The classical models for sexually transmitted infections assume homogeneous mixing either between all males and females or between certain subgroups of males and females with heterogeneous contact rates. This implies that everybody is all the time at risk of acquiring an infection. These models ignore the fact that the formation of a pair of two susceptibles renders them in a sense temporarily immune to infection as long as the partners do not separate and have no contacts with other partners. The present paper takes into account the phenomenon of pair formation by introducing explicitly a pairing rate and a separation rate. The infection transmission dynamics depends on the contact rate within a pair and the duration of a partnership. It turns out that endemic equilibria can only exist if the separation rate is sufficiently large in order to ensure the necessary number of sexual partners. The classical models are recovered if one lets the separation rate tend to infinity.

Travelling fronts in nonlinear diffusion equations

SummaryIn Fishers model for the migration of advantageous genes, in epidemic models and in the theory of combustion similar existence problems for travelling fronts and waves occur. For a general two-dimensional system of ordinary differential equations depending on a parameter the existence of trajectories connecting stationary points is established. For systems derived from diffusion problems these trajectories describe the shape of a travelling front, the corresponding value of the parameter is the propagation speed. The method allows to determine the exact value of the minimal speed in Fishers model for all interesting choices of selection parameters, i.e. for intermediate heterozygotes and for inferior heterozygotes.

Demography and epidemics

Abstract For a demographic SIRS epidemic model with vertical transmission, the balance between recruitment of new susceptibles and the persistence of the disease is investigated.

Selection models with fertility differences

SummaryWhile in the classical selection models fitness is introduced as viability or as fertility of a single genotype a model is investigated where fertility is a property of a mating type. Under this quite natural hypothesis the fundamental law of population genetics does not hold. Nevertheless for a symmetric model a thorough analysis of the equilibrium states is given. In particular it is proved that there are at most five polymorphisms. Examples can be given where all these polymorphisms occur. The stability problem can be discussed at least for the symmetric equilibria.

Pair formation models with maturation period

The standard model for pair formation is generalized to include a maturation period. This model in the form of three coupled delay equations is a special case of the general age-structured model for a two-sex population. The exact conditions for the existence of an exponential (persistent) two-sex solution are derived. It is shown that this solution is unique and locally stable. In order to achieve these results the theory of homogeneous differential equations is extended to a class of homogeneous delay equations.

Population dynamics of killing parasites which reproduce in the host

For a parasitic infection in human hosts a model is derived from basic assumptions on the population structure of the host, in particular mortality depending on age and parasite load, and on the reproduction and transmission of parasites. The model assumes the form of a system of partial differential equations. The paper contains proofs of local and global existence and existence and uniqueness of nontrivial stationary states, and a discussion of the relation to birth and death processes and other models for parasitic infections.

The discrete Rosenzweig model

Discrete time versions of the Rosenweig predator-prey model are studied by analytic and numerical methods. The interaction of the Hopf bifurcation leading to periodic orbits and the period-doubling bifurcation is investigated. It is shown that for certain choices of the parameters there is stable coexistence of both species together with a local attractor at which the prey is absent.

Model of plasmid-bearing, plasmid-free competition in the chemostat with nutrient recycling and an inhibitor

In this paper, we consider competition between plasmid-bearing and plasmid-free organisms with nutrient recycling and an inhibitor in a chemostat-type systems. We discuss the cases where the nutrient is supplied at a constant rate and the nutrient supply is time-dependent. For each case, we obtain criteria for the boundedness of solutions and persistence.

Nonhomogeneous spatial distributions of populations

SummarySpatial inhomogeneities such as nonconstant population densities usually will be attributed to random effects or to an inhomogeneous substrate. Such an explanation may be incorrect since from certain chemical reactions it is known that the interaction of species together with diffusion may generate nonhomogeneous spatial structures. However, the effect of boundary conditions has been so far neglected. In this paper nonlinear and linear interaction-diffusion models are investigated under various side-conditions by analytic methods and by computer simulations. A remarkable fact, as compared with earlier results in the field, is the example of an interaction-diffusion process which in the whole space has only the constant as a stable limit distribution, whereas the introduction of a side condition, e.g. a population reservoir or a barrier, leads to standing spatial population waves.