Predator-Prey Systems

The BZ reaction above is an example of a system in which different types of components (reactants and products) are connected in a web of interactions. The couplings of components, in particular the feedback effects, means that the state of each component becomes strongly dependent on the state of the others. Such systems can be modelled by two approaches: agent based modelling or the phenomenological approach. In agent based modelling one treats each individual component in the system as a basic unit (agent) which interacts in a well-defined way with other agents or the environment. We will encounter some examples of agent based modelling in the next chapter. In this section we will use the phenomenological approach in which some macroscopic variables are chosen to represent the state of the system. For example, when we studied the Verhulst model of population growth earlier, a single variable $P_n$, the population number, was used to characterise a species of organisms. An equation was obtained for the evolution of $P_n$ which contained some undetermined parameters. That is the phenomenological approach. (We could instead also have tried to treat each individual member of the species as a basic unit and modelled the interactions among the individuals, leading to an agent based model.) Let us now study some phenomenological equations describing the competition between two species. Volterra and Lotka proposed the mathematical equations below to describe the observed cyclic variation in the population of two species of fish: Big fish population denoted by $B$ and little fish population denoted by $L$. The equations are

$${ d L \over d t} = L (a-bB) ,$$ (5.5)

$${ d B \over d t} = B (cL-d) ,$$ (5.6)

where $a,b,c,d$ are positive constants which can be given the following interpretation. The parameter $a$ represents the reproduction rate of the little fish, while $b$ is a measure of the likelihood that a big fish eat will eat a little fish. Thus the first equation(5.5) describes how the potential exponential growth of the little fish population is moderated by a predator. Similarly, the parameter $d$ in the second equation describes the rate of decrease (death) of the big fish population in the absence of food (little fish), while the paramrter $c$ is a measure of the rate at which the big fish population increases by feeding on the little fish. The coupled differential equations can be solved numerically and typically display limit cycles (periodic solutions). This is explored further in the exercises. It is easy to see that the equations have a stationary solution (that is, a time-independent solution or fixed point) at $B^{*} = a/b$ and $L^{*} = d/c$. This solution has some counterintuitive features which highlights the often non-obvious and intricate nature of nonlinear dynamic systems. For example, suppose one wanted to increase the equilibrium population of little fish. Naively one might think of increasing the little fish reproductive rate $a$ to achieve this, but as we see from the solution $L^{*} = d/c$ that this is incorrect! Generalisations of the Lotka-Volterra can be used to describe the spread of diseases. See for example, Ref.[4] for a brief description of some epidemic models and Ref.[11] for some generalisations of the Lotka-Volterra approach.