📂Dynamics

Food Chain System in Dynamics

Model ¹

$$\begin{align*} \dot{R} =& R \left( 1 - {\frac{ R }{ K }} \right) - {\frac{ x_{c} y_{c} C R }{ R + R_{0} }} \\ \dot{C} =& x_{c} C \left( {\frac{ y_{c} R }{ R + R_{0} }} - 1 \right) - {\frac{ x_{p} y_{p} P C }{ C + C_{0} }} \\ \dot{P} =& x_{p} P \left( {\frac{ y_{p} C }{ C + C_{0} }} - 1 \right) \end{align*}$$

Variables

• $R(t)$: The density of resource at time $t$.
• $C(t)$: The density of consumer at time $t$.
• $P(t)$: The density of predator at time $t$.

Parameters

• $K = 0.94$: The carrying capacity of the resource.
• $x_{c} = 0.4$, $y_{c} = 1.7$, $R_{0} = 0.16129$: Values related to the consumers preying on the resource.
• $y_{p} = 5.0$, $x_{p} = 0.08$, $C_{0} = 0.5$: Values related to the predators preying on the consumers.

Description

The introduced $3$-dimensional food-chain system follows a logistic growth model for resources, and can be viewed as a coupled dynamic system of two Lotka-Volterra predator-prey models: the resource-consumer and the consumer-predator.

Typically, systems composed of such simple structures cannot be complex, but with the inclusion of a Holling Type II functional response between $R-C-P$, it becomes a chaotic system. Indeed, the bifurcation diagrams drawn by varying $K, y_{c}, y_{p}$ are as follows.

From a research standpoint, understanding such systems could add value to a paper. Although there are other examples of $3$-dimensional chaotic systems, such as the Lorenz attractor, it is too famous and overused to hold appeal as a benchmark. Moreover, as of now, it may seem less complex since it can be expressed by polynomial functions within $2$ dimensions. In contrast, the food-chain system inherently carries intuitive significance in ecology, is chaotic, and includes rational functions in its formulation, making it relatively complex ².