Simple models for studying complex spatiotemporal patterns of animal behavior

Minimal mathematical models able to explain complex patterns of animal behavior are essential parts of simulation systems describing large-scale spatiotemporal dynamics of trophic communities, particularly those with wide-ranging species, such as occur in pelagic environments. We present results obtained with three different modelling approaches: (i) an individual-based model of animal spatial behavior; (ii) a continuous taxis-diffusion-reaction system of partial-difference equations; (iii) a 'hybrid' approach combining the individual-based algorithm of organism movements with explicit description of decay and diffusion of the movement stimuli. Though the models are based on extremely simple rules, they all allow description of spatial movements of animals in a predator-prey system within a closed habitat, reproducing some typical patterns of the pursuit-evasion behavior observed in natural populations. In all three models, at each spatial position the animal movements are determined by local conditions only, so the pattern of collective behavior emerges due to self-organization. The movement velocities of animals are proportional to the density gradients of specific cues emitted by individuals of the antagonistic species (pheromones, exometabolites or mechanical waves of the media, e.g., sound). These cues play a role of taxis stimuli: prey attract predators, while predators repel prey. Depending on the nature and the properties of the movement stimulus we propose using either a simplified individual-based model, a continuous taxis pursuit-evasion system, or a little more detailed 'hybrid' approach that combines simulation of the individual movements with the continuous model describing diffusion and decay of the stimuli in an explicit way. These can be used to improve movement models for many species, including large marine predators.