A nonlocal kinetic model for predator–prey interactions

Abstract

We extend the aggregation model from Fetecau (2011) by adding a field of vision to individuals and by including a second species. The two species, assumed to have a predator–prey relationship, have dynamics governed by nonlocal kinetic equations that include advection and turning. The latter is the main mechanism for aggregation and orientation, which results from interactions among individuals of the same species as well as predator–prey relationships. We illustrate numerically a diverse set of predator–prey behaviors that can be captured by this model. We show that a prey’s escape outcome depends on the social interactions between its group members, the prey’s field of vision and the sophistication of the predator’s hunting strategies.

Appendix: Modelling the probability functions in equations (13)–(15)

We first describe the modeling of w _a . Since w _a is a probability function, we have

$$ \int_{-\pi}^{\pi} w_a \bigl( \phi' -\phi, \phi'-\psi \bigr) \,d\phi= 1. $$

(35)

The probability function w _a is modeled by

$$ w_a \bigl( \phi' -\phi, \phi'-\psi \bigr) = g_{\sigma_a} \bigl( \phi' -\phi- v_a \bigl( \phi'-\psi \bigr) \bigr), $$

(36)

where $g_{\sigma_{a}}$ is an approximation of the delta function with width σ _a , and v _a is a turning function. The decision making individual can turn to any direction within a specific range. This range is centered around the direction

$$\phi= \phi' - v_a \bigl( \phi'-\psi \bigr), $$

which happens when the argument of the function $g_{\sigma_{a}}$ is zero. The parameter σ _a >0 measures the width of the turning range the decision making individual will move into—see Fig. 3(b). The smaller the σ _a , the more accurate the turning. If σ _a is large, then the range is wide and the decision making individual can move anywhere within the range.

The function to describe $g_{\sigma_{a}}$ is taken to be

$$ g_{\sigma_a} ( \eta ) = \frac{1}{\sqrt{\pi}\sigma_a} \sum _{z \in\mathbb{Z}}e^{- (\frac{\eta+ 2\pi z}{\sigma_a} )^2}, $$

(37)

which is a periodic Gaussian with extra contributions from full rotations.

The turning function v _a is modeled as

$$ v_a ( \eta ) = k_a \sin\eta, $$

(38)

where k _a is a constant between 0 and 1 that describes how much the decision making individual will turn due to attraction (see Fig. 3(b)).

An alternative choice is v _a (η)=k _a η, which may be more biologically realistic. However the choice (38) for v _a is made because it is periodic and works well with the fast Fourier transform, which the numerics in this paper is based on, as discussed in Sect. 3.

The probability functions w _r and w _al are defined through the same steps as equations (36)–(38), using approximations $g_{\sigma_{r}}$, $g_{\sigma_{\mathit{al}}}$ to the delta function, with widths σ _r , σ _al , turning functions v _r , v _al , and turning strengths k _r , k _al , respectively.

There is a major difference between w _a and w _r which has to be pointed out. Namely, in the definition of w _r , k _r must be between −1 and 0 instead of between 0 and 1, as is k _a . This is the only place where the negative factor comes in to enforce the negative behavior of the repulsion interaction. The attraction and alignment interactions are positive because they respond positively to the surrounding individuals.