We consider patch-based population dynamics, called metapopulations, and investigate the spread of two species and their competition for habitat areas. Let 0 ≤ p1,p2 denote the proportion (density) of the areas occupied by the species 1 and 2, respectively. Obviously, 0 ≤ p1 + p2 ≤ 1. Both species can locally extinct and colonize empty patches or patches occupied by the other with different rates. Hence the model is P1′ = κ1P1 (1 - P1 - P2) - e1p1p2-C1P1P2 - C2P1P2 P2′ = κ1P1 (1 - P1 - P2) - e1p1p2-C1P1P2 - C2P1P2 where ej are the extinction rates, κ1 are the rates of colonization of empty patches, and species "i" overcolonize "j" with the rate c1(i,j=1,2; i≠j) We investigate the asymptotic stability properties of the equilibria of such systems and we give a complete characterization of the parameter space. We prove that there can appear a globally asymptotically stable coexisting equilibrium. On the other hand, this equilibrium can become unstable and then the exclusive equilibria are only locally asymptotically stable. We have also developed dynamic applications in Wolfram Mathematica to illustrate our results.

We used cellular automata models to investigate the effect of initial pattern geometry on competition. We measured the average proportion of sites with foreign neighbours to track interspecific segregation during pattern development. Our simulation results show that intraspecific aggregation can considerably slow down the extinction of the weaker competitor. A series of experiments was performed to estimate the expected time to extinction for the weaker species. The perimeter-to-area ratio of the initial configuration proved to be an adequate determinant of expected time to-extinction. Furthermore, we demonstrated that the degree of aggregation is closely related to the local density dependence of the colonization functions.

In this paper we summarize our concepts and practice on computer-aided mathematical experimentation, and illustrate them by Mathematica projects that we have developed for our research and the courses "Computer-aided mathematical modelling" and "Computer Algebra I-II" held for students of life sciences at University of Szeged and computational engineering at TFH Berlin, University of Applied Sciences.