8 Lotka-Volterra system with delays

The system of two differential equations with delays

$$\left\{\vphantom{ \begin{array}{l} \vspace{2mm} \displaystyle \do... ...-\tau}^{0} F_2(s) x_1 (t+s) ds \biggr] x_2 (t) ,\end{array}}\right. \begin{array}{l} \vspace{2mm} \displaystyle \dot x_1 (t) = ... ...limits_{-\tau}^{0} F_2(s) x_1 (t+s) ds \biggr] x_2 (t) ,\end{array}$$ (8.1)

is the Lotka-Volterra model describing the population dynamics of two competing species (prey-predation population model).

The conditional representation of (8.1) is

$$\left\{\vphantom{ \begin{array}{l} \vspace{2mm} \displaystyle \do... ...mits_{-\tau}^{0} F_2(s) y_1 (s) ds \biggr] x_2 ,\end{array}}\right. \begin{array}{l} \vspace{2mm} \displaystyle \dot x_1 = \big... ... \int\limits_{-\tau}^{0} F_2(s) y_1 (s) ds \biggr] x_2 ,\end{array}$$ (8.2)

so to simulate system (8.2) it is necessary to set the following parameters:

- constants $\varepsilon$, $\gamma_{1}^{}$ and $\gamma_{2}^{}$,

- functions F₁( ^. ), F₂( ^. ),

- an initial point x⁰,
- an initial prehistory y⁰( ^. ) = {y⁰(s), - $\tau$ $\leq$ s < 0}.

One can simulate system (8.2) using the function lv45.