Consider the predator-prey models
\[ dx = [a x - b x^2 - c_1 \varphi (x,y) x] dt + \alpha x dW_1, \]
\[ dy = [r y - c_2 \varphi (x,y) y] dt + \beta y dW_2, \]
where \(x\) and \(y\) represent the population densities of prey and predator at time \(t \geq 0\); \(a\) and \(r\), \(b\), and \(c_i\) are positive constants of prey and predator intrinsic growth rates, death rate measuring the strength of competition among the prey individuals and conversion rates of the predator-prey model, respectively. Moreover, \(\varphi (x,y) \) models the functional response of the predator to the prey density per unit time.
In this paper, the authors are interested in the study of the case of modified Leslie-Gower and Holling-type-II functionals of the form
\[ \varphi (x,y) = y / (m+x), \]
where \(m\geq 0\) measures the extent to which the environment provides protection to the prey \(x\). Now, suppose these models are perturbed by independent standard Wiener processes \(W_i\) in the parameters \(a\) and \(b\) with variance parameters \(\alpha\) and \(\beta\). Thus, one arrives at nonlinear stochastic differential equations with linear multiplicative noise and quadratic nonlinearities.
First, the authors prove the existence of unique, positive global solutions to these systems with positive initial values \(x_0, y_0 > 0\). Second, they study its long-time behavior. Third, the system is investigated with respect to extinction and persistence conditions. As their major proof technique, the authors apply well-known comparison theorems on upper and lower solution bounds and the knowledge on explicit solutions of stochastic logistic equations with linear multiplicative white noise in \(\mathbb{R}^1\). Finally, standard Milstein methods are used for some numerical simulations to illustrate their findings.