Modelling of spatial spreading of invasions in the discrete homogeneous environment

A. Yu. Perevaryukha

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 23 (2020) no. 1, pp. 44-67.

Résumé

We consider the problem of modeling important processes of biological invasions in discrete space using a new algorithm for reproduction and death of the constituent populations of individual cells. The inclusion of $x(t-\tau)$ in the continuous model of the delay is an obvious way to diversify the options for the behavior of the trajectory, but without expanding the structure and increasing the dimension of the phase space. Using population models with a deviating argument $\dot x =rf(x-\tau)- \Psi(x^k(t-\nu))$ in some cases does not follow the realities of life. An explicit form of delay is suitable for inclusion in the phenomenological models of rapidly ripening species. Another spatial model is relevant, where time factors can be set visually. The goal of our work is to investigate the algorithm for transforming the state of cells in the space of a square lattice and to obtain the unsteady dynamics of clusters of two populations with an explicit interpretation of the parameters of time delay. To demonstrate the development of invasion with a complex of realistic factors of the temporary aftereffect, we propose a cellular automaton algorithm. Our algorithm is not yet another modification of “Life” or “Aqua-Tor”, since the neighborhood with eight adjacent points and three colors of cells in a square lattice are used. For the lag phenomenon in the new algorithm, the following parameters are responsible: restrictions on the rate of reproduction of individuals and updating the environment, and also the time of migration of new individuals to their resources in space. We carry out a computational implementation of the transformation of the given initial state of cells during invasion in accordance with the transformation rules. The scenario for the cyclicality of the two main quantities in the system is shown. The occurrence or destruction of cycles depends on the rate of renewal of green cells. The forms of transformation of the state of cells confirm that the delay that we formalize in the Nicholson model is much more relevant to the dynamics of the interaction of the invader species and the environment that supports its existence. The effect of the delay of the value $\tau $ does not make sense in modeling when identifying it with the characteristic of a directly biological species. When developing a response from the environment to invasion, the lag of the value $\nu $ is different in essence than when restoring resources. The practical significance of our work lies in modeling the movement of the crest of the invasive wave and the final synchronization of vibration peaks in such warring predator species as Beroe ovata and Mnemiopsis leidyi in the Black and Azov Seas an ecological system in the form of "predator $A\Longleftrightarrow$ predator $B$". The oscillatory behavior of two populations differs from the scenarios that can be obtained in continuous models.

Détail

Keywords: Beroe ovata, algorithmic models, delay in process models, critical scenarios of population dynamics, regulation mechanisms, synchronization of predators oscillations, Beroe ovata, infections spreading.
Mots-clés : population invasion fronts