Jozsef Zoltan Farkas

Structured and Unstructured Continuous Models for Wolbachia Infections

We introduce and investigate a series of models for an infection of a diplodiploid host species by the bacterial endosymbiont Wolbachia. The continuous models are characterized by partial vertical transmission, cytoplasmic incompatibility and fitness costs associated with the infection. A particular aspect of interest is competitions between mutually incompatible strains. We further introduce an age-structured model that takes into account different fertility and mortality rates at different stages of the life cycle of the individuals. With only a few parameters, the ordinary differential equation models exhibit already interesting dynamics and can be used to predict criteria under which a strain of bacteria is able to invade a population. Interestingly, but not surprisingly, the age-structured model shows significant differences concerning the existence and stability of equilibrium solutions compared to the unstructured model.

Linear stability and positivity results for a generalized size-structured Daphnia model with inflow§

We employ semigroup and spectral methods to analyze the linear stability of positive stationary solutions of a generalized size-structured Daphnia model. Using the regularity properties of the governing semigroup, we are able to formulate a general stability condition, which permits an intuitively clear interpretation in a special case of model ingredients. Moreover, we derive a comprehensive instability criterion that reduces to an elegant instability condition for the classical Daphnia population model in terms of the inherent net reproduction rate of Daphnia individuals. §Dedicated to Professor Miklós Farkas on the occasion of his 75th birthday.

Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback

In this work we consider a size-structured cannibalism model with the model ingredients (fertility, growth, and mortality rate) depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system, in particular on the effect of cannibalism on the long-term dynamics. To this end, we formally linearize the system about steady state and establish conditions in terms of the model ingredients which yield uniform exponential stability of the governing linear semigroup. We also show how the point spectrum of the linearized semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup.

Semigroup analysis of structured parasite populations

Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a pop- ulation is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicon- traction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.

Steady states in a structured epidemic model with Wentzell boundary condition

We introduce a non-linear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment that carries mass; hence the model is equipped with generalized Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for e.g., Wolbachia in a mosquito population. Therefore, the (infinite dimensional) non-linearity arises in the recruitment term. First, we establish global existence of solutions and the principle of linearised stability for our model. Then, in our main result, we formulate simple conditions which guarantee the existence of non-trivial steady states of the model. Our method utilises an operator theoretic framework combined with a fixed-point approach. Finally in the last section, we establish a sufficient condition for the local asymptotic stability of the positive steady state.

Physiologically structured populations with diffusion and dynamic boundary conditions.

We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. We equip the model with generalized Wentzell-Robin (or dynamic) boundary conditions. This approach allows the modelling of populations in which individuals may have distinguished physiological states. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. These results are obtained by establishing dissipativity of a suitably perturbed semigroup generator. We also show that solutions of the model exhibit balanced exponential growth, that is, our model admits a finite-dimensional global attractor. In case of strictly positive fertility we are able to establish that solutions in fact exhibit asynchronous exponential growth.

On a size-structured two-phase population model with infinite states-at-birth

In this work, we introduce and analyze a linear size-structured population model with infinite states-at-birth. We model the dynamics of a population in which individuals have two distinct life-stages: an “active” phase when individuals grow, reproduce and die and a second “resting” phase when individuals only grow. Transition between these two phases depends on individuals’ size. First we show that the problem is governed by a positive quasicontractive semigroup on the biologically relevant state space. Then, we investigate, in the framework of the spectral theory of linear operators, the asymptotic behavior of solutions of the model. We prove that the associated semigroup has, under biologically plausible assumptions, the property of asynchronous exponential growth.

ON THE LINEARIZED STABILITY OF AGE-STRUCTURED MULTISPECIES POPULATIONS

We consider a general nonlinear age-structured population model with n interacting species. We deduce the characteristic function in the form of a determinant of an n-by-n matrix. Then we formulate some biologically meaningful sufficient conditions for the stability (resp., instability) of positive stationary solutions of the system.

On the net reproduction rate of continuous structured populations with distributed states at birth

We consider a nonlinear structured population model with a distributed recruitment term. The question of the existence of non-trivial steady states can be treated (at least) in three different ways. One approach is to study spectral properties of a parametrised family of unbounded operators. The alternative approach, which we develop here, is based on the reformulation of the partial differential equation as an integral equation. In this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity. Finally, we discuss a third approach, which is based on a finite rank approximation of the recruitment operator.

Steady states in hierarchical structured populations with distributed states at birth

We investigate steady states of a quasilinear first order hyperbolic partial integro-differential equation. The model describes the evolution of a hierarchical structured population with distributed states at birth. Hierarchical size-structured models describe the dynamics of populations when individuals experience size-specific environment. This is the case for example in a population where individuals exhibit cannibalistic behavior and the chance to become prey (or to attack) depends on the individuals size. The other distinctive feature of the model is that individuals are recruited into the population at arbitrary size. This amounts to an infinite rank integral operator describing the recruitment process. First we establish conditions for the existence of a positive steady state of the model. Our method uses a fixed point result of nonlinear maps in conical shells of Banach spaces. Then we study stability properties of steady states for the special case of a separable growth rate using results from the theory of positive operators on Banach lattices.