Dariusz Wrzosek

Long-time behaviour of solutions to a chemotaxis model with volume-filling effect

The Lyapunov functional is constructed for a quasilinear parabolic system which models chemotaxis and takes into account a volume-filling effect. For some typical case it is proved that the ω -limit set of any trajectory consists of regular stationary solutions. Some lower and upper bounds on the stationary solutions are found. For a given range of parameters there are stationary solutions which are inhomogeneous in space.

MATHEMATICAL MODELLING OF CANCER INVASION: THE IMPORTANCE OF CELL–CELL ADHESION AND CELL–MATRIX ADHESION

The process of invasion of tissue by cancer cells is crucial for metastasis — the formation of secondary tumours — which is the main cause of mortality in patients with cancer. In the invasion process itself, adhesion, both cell–cell and cell–matrix, plays an extremely important role. In this paper, a mathematical model of cancer cell invasion of the extracellular matrix is developed by incorporating cell–cell adhesion as well as cell–matrix adhesion into the model. Considering the interactions between cancer cells, extracellular matrix and matrix degrading enzymes, the model consists of a system of reaction–diffusion partial integro–differential equations, with nonlocal (integral) terms describing the adhesive interactions between cancer cells and the host tissue, i.e. cell–cell adhesion and cell–matrix adhesion. Having formulated the model, we prove the existence and uniqueness of global in time classical solutions which are uniformly bounded. Then, using computational simulations, we investigate the effects of the relative importance of cell–cell adhesion and cell–matrix adhesion on the invasion process. In particular, we examine the roles of cell–cell adhesion and cell–matrix adhesion in generating heterogeneous spatio-temporal solutions. Finally, in the discussion section, concluding remarks are made and open problems are indicated.

A Chemotaxis Model with Threshold Density and Degenerate Diffusion

A quasilinear degenerate parabolic system modelling the chemotactic movement of cells is studied. The system under consideration has a similar structure as the classical Keller-Segel model, but with the following features: there is a threshold value which the density of cells cannot exceed and the flux of cells vanishes when the density of cells reaches this threshold value. Existence and uniqueness of weak solutions are proved. In the one-dimensional case, flat-hump-shaped stationary solutions are constructed.

Predation‐Mediated Coexistence of Large‐ and Small‐Bodied Daphnia at Different Food Levels

Using an individual‐based age‐structured population model (a combination of O’Brien’s apparent‐prey‐size approach, Eggers’s reactive‐field‐volume model, and Holling’s disk equation), we could predict that (1) a Daphnia population could be kept at low density by fish predation irrespective of food level, with greater recruitment at higher food being instantly compensated for by raised mortality reflecting increased predation, and (2) Daphnia density levels are species specific and inversely related to both body size at first reproduction and the reaction distance at which a foraging fish sees its Daphnia prey. These two hypotheses were experimentally tested in outdoor mesocosms with two Daphnia species of different body sizes grown in the absence or presence of fish that were allowed to feed for 2–3 h each evening. While each Daphnia quickly reached high density with reproduction halted by food limitation in the absence of fish, the populations stayed at much lower species‐specific density levels, similar in low and high food concentrations, in the presence of fish. This suggests that our model offers a reasonable mechanistic explanation for the coexistence of large‐ and small‐bodied zooplankton in proportions reflecting their body sizes throughout habitats comprising a wide productivity spectrum, with each species at a density level at which it becomes included in a predator’s diet.

Limit cycles in predator-prey models

The general model of interaction between one predator and one prey is studied. A unimodal function of rate of growth of the prey and concave down functional response of the predator is assumed. In this work it is shown that for a given natural number n there exist models possessing at least 2n + 1 limit cycles. It is also proved, applying the Hopf bifurcation theorem, that a model exists with a logistic growth rate of the prey and concave down functional response that has at least two limit cycles.

Global regularity versus infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion

A system of quasi-linear parabolic and elliptic-parabolic equations describing chemotaxis is studied. Due to the assumed presence of a volume-filling effect it is assumed that there is an impassable threshold for the density of cells. This assumption leads to singular or degenerate operators in both the diffusive and the chemotactic components of the flux of cells. We improve results from earlier works and find critical conditions which reflect the interplay between diffusion and chemotaxis and warrant that classical solutions are global in time and separated uniformly from the threshold. In the case of degenerate diffusion for the elliptic-parabolic version of the model we prove the existence of radially symmetric solutions which exhibit a phenomenon of infinite-time singularity formation in that they are global and smooth but attain the threshold in the large time limit.

On the oort-hulst-safronov coagulation equation and its relation to the Smoluchowski equation

A connection is established between the classical Smoluchowski continuous coagulation equation and the Oort--Hulst--Safronov coagulation equation via generalized coagulation equations. Existence of solutions to the Oort--Hulst--Safronov coagulation equation is shown, and the large time behavior and the occurrence of gelation are studied as well. It is also shown that a compactly supported initial distribution propagates with finite speed.

Singularity formation in chemotaxis systems with volume-filling effect

A parabolic–elliptic model of chemotaxis which takes into account volumefilling effects is considered under the assumption that there is an a priori threshold for the cell density. For a wide range of nonlinear diffusion operators including singular and degenerate ones it is proved that if the taxis force is strong enough with respect to diffusion and the initial data are chosen properly then there exists a classical solution which reaches the threshold at the maximal time of its existence, no matter whether the latter is finite or infinite. Moreover, we prove that the threshold may even be reached in finite time provided the diffusion of cells is non-degenerate.

The Discrete Coagulation Equations with Collisional Breakage

The discrete coagulation equations with collisional breakage describe the dynamics of cluster growth when clusters undergo binary collisions resulting either in coalescence or breakup with possible transfer of matter. Each of these two events may happen with an a priori prescribed probability depending for instance on the sizes of the colliding clusters. We study the existence, density conservation and uniqueness of solutions. We also consider the large time behaviour and discuss the possibility of the occurrence of gelation in some particular cases.

Predator–prey model with diffusion and indirect prey-taxis

We analyze predator–prey models in which the movement of predator searching for prey is the superposition of random dispersal and taxis directed toward the gradient of concentration of some chemical released by prey (e.g. pheromone), Model II, or released from damaged or injured prey due to predation (e.g. blood), Model I. The logistic O.D.E. describing the dynamics of prey population is coupled to a fully parabolic chemotaxis system describing the dispersion of chemoattractant and predator’s behavior. Global-in-time solutions are proved in any space dimension and stability of homogeneous steady states is shown by linearization for a range of parameters. For space dimension N ≤ 2 the basin of attraction of such a steady state is characterized by means of nonlinear analysis under some structural assumptions. In contrast to Model II, Model I possesses spatially inhomogeneous steady states at least in the case N = 1.