Stochastic modelling of Karlotoxin influence on prey

Abstract

Abstract Karlodinium veneficum is type of dinoflagellate plankton present in coastal regions. Harmful algae blooms resulting from uncontrolled growth of K. veneficum often leads to fish kills. They release a toxin named Karlotoxin that is known to affect their prey’s bio-locomotion by stunning and slowing them down. We investigate whether the toxin plays a role in aggregating the prey around the a predator, thereby leading to a local increase in prey density. To achieve this, we closely examine the toxin’s influence on the prey’s probability density distribution with different assumptions on their relative speed in 1D, with either the predator being stationary or swimming at a constant speed. When the predator is stationary, we fully solve the prey’s density distribution for all times, and verify the result by a Monte-Carlo simulation. For a swimming predator, we find the steady-state density distribution of prey analytically. When the predator’s speed $s$ is strictly greater (or less) than the prey ($s-1>0$ or $s-1<0$), the results are verified by Monte-Carlo simulations; when their relative speed $s-1$ has roots, we use the Frobenius method to perform a local analysis for the prey’s density at steady state near the roots, and use the result to derive a scheme for finding the analytical solution. This solution is then verified numerically using a finite difference method. When the roots $x_{1}$ and $x_{3}$ satisfy $s^{\prime}(x_{1})<0$ and $s^{\prime}(x_{3})>0$, we show that the probability density for the prey has a form $|x-x_{1}|^{-s^{\prime}(x_{1})^{-1} -1}$ near the root $x_{1}$, leading to either an integrable singularity or a local maximum. Near the root $x_{3}$, the prey’s density can be represented as a Taylor series and is smooth. In most of the cases mentioned above, toxin leads to the aggregation of prey, however the maximum density does not always occur where the toxin has the highest concentration.