**2. Mathematical Systems**

In this paper, we introduce non-dimensional systems of two and three trophics. Holling type I and II functional and numerical responses are used to describe the predation of predators on prey and the effect of prey consumption on predators. Holling type I represents a linear function, whereas Holling type II represents a nonlinear function. The model can be formulated as:

The system of two trophics is as follows:

$$\begin{aligned} \frac{dx}{dt} &= \mathbf{x}\left(1 - \frac{\mathbf{x}}{k}\right) - f(\mathbf{x})y, \\\\ \frac{dy}{dt} &= -\mu y + R\_1 y \left(1 - \frac{y}{k\_y}\right), \end{aligned} \tag{1}$$

with initial conditions

$$\mathbf{x}(0) = \mathbf{x}\_{0\prime} \text{ (0) } = y\_0 \text{.}$$

The system of three trophics is as follows:

$$\begin{aligned} \frac{dx}{dt} &= x\left(1 - \frac{x}{k}\right) - f(x)y - g(x)z = xf(x, y, z), \\\\ \frac{dy}{dt} &= -uy + R\_1y\left(1 - \frac{y}{k\_y}\right) - c\_1yz = yL\_1(x, y, z), \\\\ \frac{dz}{dt} &= -wz + R\_2z\left(1 - \frac{z}{k\_z}\right) - c\_2yz = zL\_2(x, y, z), \end{aligned} \tag{2}$$

with initial conditions

$$\mathbf{x}(0) = \mathbf{x}\_{0\prime} \ (0) = y\_{0\prime} \ z(0) = z\_0.$$

The different parameters in systems (1) and (2) are explained as follows. The intrinsic growth rate of prey is 1. In the case of Holling type I, *f*(*x*) = *αx* and *g*(*x*) = *βx* are the functional responses to predators *y* and *z*, respectively, whereas in the case of Holling type II, *f*(*x*) = *αx* 1+*h*1*α<sup>x</sup>* and *g*(*x*) = *βx* <sup>1</sup>+*h*2*β<sup>z</sup>* are the functional responses to predators *y* and *z*, respectively. For type I, the numerical responses are *R*1 = *e*1*αx* and *R*2 = *<sup>e</sup>*2*β<sup>x</sup>* of the predators *y* and *z*, respectively. For Holling type II, *R*1 = *e*1*αx* 1+*h*1*α<sup>x</sup>* and *R*2 = *<sup>e</sup>*2*β<sup>x</sup>* <sup>1</sup>+*h*2*β<sup>z</sup>* . The parameters *α* and *β* measure the efficiency of the search and the capture of predators *y* and *z*, respectively. In the absence of prey *x*, the constants *u* and *w* are the death rates of predators *y* and *z*, respectively. *h*1 and *h*2 represent the handling and digestion rates of the predators, respectively, and *e*1 and *e*2 symbolize the efficiency of converting consumed prey into predator births. The carrying capacities *ky* = *a*1*x* and *kz* = *a*2*x* are proportional to the available amount of prey. In this paper, we assume *a*1 = *a*2 = 1 to simplify the mathematical analysis. *c*1 and *c*2 measure the interspecific competition between the predators. All the parameters and initial conditions of systems (1) and (2) are assumed to be positive values.
