We present two finite-difference algorithms for studying the dynamics of spatially extended predator-prey interactions with the Holling type II functional response and logistic growth of the prey. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. This is advantageous as it is well-known that the dynamics of approximations of differential equations (DEs) can differ significantly from that of the underlying DEs themselves. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. Furthermore, there are implementational advantages of the methods. For example, due to the structure of the resulting linear systems, standard direct, and iterative solvers are guaranteed to converge. We also present the results of numerical experiments in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs MATLAB: . Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/, to investigate the key dynamical properties of spatially extended predator-prey interactions.

We study the numerical approximation of the solutions of a class of nonlinear reaction-diffusion systems modelling predator-prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction-diffusion system (Garvie and Blowey in Eur J Appl Math 16(5):621-646, 2005). Mathematics Subject Classification (2000)65M60 · 65M15 · 65M12 · 92D25 · 35K55 · 35K57

a b s t r a c tWe present a new algorithm for estimating parameters in reaction-diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer-Meinhardt reaction-diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available.

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We examined the dispersal of larvae (glochidia) of a common unionid mussel species, Actinonaias ligamentina, which need to attach to a host fish in order to develop into juveniles, in a lowland river (Sydenham River, Ontario, Canada). Generally, the decline in the number of glochidia captured with distance from release was best described by an inverse power function. The highest proportion was found in the first net 4 m downstream (range 0.1-3.6%), but a small proportion of glochidia was captured 96 m downstream (0-0.03%). This indicates that infestation of host fish may occur several tens to hundreds of meters downstream of the adults' location, even at relatively low flow conditions (mean velocity, 15 cm s 21 ). Dispersal distances increased with velocity, but the number of glochidia sampled at a given location can vary considerably due to stochastic effects of turbulence, especially at shorter distances. Individual trials could, therefore, deviate considerably from the predictions of an existing turbulent transport model (local exchange model), but overall there was a good correlation between measured data and model prediction. However, model predictions were quantitatively much higher than measured values (i.e., . 50 fold in some cases), which could be in part due to several simplifying assumptions of the model. * Corresponding