Population Dynamics (Prey- Predator Relationship)

The concepts of natural growth and the prey- predator relationship is fundamental to achieving an appreciation of population sustainability.

In the late 1960’s, not long after I had completed a course on chemical reaction rates, I attended a lecture delivered by a young Professor Robert May on the prey- predator relationship. This insight into the parameters controlling the changes in populations of prey and predators, together with the conditions leading to sustainable populations was both fascinating and memorable.

Without predators or other complicating factors, the most basic natural law is the exponential growth or decay of a population with time. Accelerating growth occurs when the birth rate is greater than the death rate or, alternatively an exponential decay is found when the death rate exceeds the birth rate. This is shown graphically in Figure 1 below.

For those who are comfortable with applied mathematics, the exponential relationship is written:

N = No *exp(k*t)

where N is the population at time t, No is the initial population and k, represents birth rate minus death rate. This exponential relationship is derived by the solution of a differential equation which states that the rate of change of population (dN/dt) is proportional to the population, N at time, t, where the constant of proportionality, k, for biological systems is the birth rate minus the death rate. When the birth rate dominates over the death rate, the constant k is positive and when k is negative, the death rate dominates. The graph of this exponential relationship shown in Figure 1 and indicates the effect of differences between birth and death rates for five arbitrary cases.





















































Figure 1 Graph of Population (%) against Time (arbitrary units) showing the effect of birth rate dominating over death rate and death rate dominating over birth rate.

It is clear that the population eventually explodes when the birth rate dominates and the population eventually dies out when the death rate dominates. Only in balance is the population sustainable.

Prey- Predator relationships

The physical and financial world has many examples of the prey existing in the presence of predators. These systems can show a range of outcomes of population over time which include stable population, oscillations, decline or run away growth. There is a feedback and control mechanism at work in these systems, as the rate of change of prey population in an environment of unrestricted food resources, is not only proportional to the population of prey, but also the predation rate. Similarly, the rate of change of predator population is not only proportional to the predator population, but moderated by the predation rate of the prey available as a food source.

This is widely known as the Lotka- Voltera model which arose from a theory of oscillating chemical reactions. The model uses simplifying assumptions that do not bear close scrutiny in the real world, but does have value in providing insight into population dynamics as parameters are changed. These parameters are the ratio of initial populations of prey and predator, together with their birth rates, death rates and predation rate. Two coupled differential rate equations for prey and predators are solved simultaneously to arrive at the population of both prey and predators as a function of time. The model is usually exercised to show the case of oscillatory behaviour in both prey and predator populations for a particular set of parameters and reflects population oscillations observed in some prey- predator populations in nature. We can also point to economic systems where boom and bust cycles are observed and conclude that similar models may provide insights into these oscillatory phenomena. Other initial conditions and parameters can be selected to simulate the demise or the out of control growth of prey or predator populations. The equations and the feedback conditions for stability are reminiscent of an Engineers analysis of stability in a mechanical, chemical or electronic system.

Prediction of Collapse in a Civilisation.

The Human and Nature Dynamical model (HANDY) is a more complex predator- prey model that enables important social insights through humans playing the role of predators and nature playing the role of the resources preyed upon. Population, “EcoDollars” and time are three dimensions used in the model. The population is divided into two components: (1) “commoners” (workers), who both generate wealth and consume resources and (2) “elites”, who consume resources. The “EcoDollars” are also subdivided into two components: (3) “nature”, which supplies “EcoDollars” and (4) “wealth”, which allows accumulation of “EcoDollars”. Four simultaneous differential equations connect these four time dependent variables. One set of realistic parameters was selected and used in three scenarios. This led to the conclusion that an egalitarian society (no “elites”) and an equitable society (“elites” and “commoners” with equal consumption) could “soft land” to a stable population equilibrium provided the resource depletion ratio was sufficiently restrained. However as the resource depletion ratio is increased, oscillation in population and cycles of prosperity and collapse are observed. Increasing the inequality factor for “commoners” and “elites” to 100, led to a complete collapse of both populations and the economy, despite the imposition of an initial ratio of “elites” to “commoners” of 0.2 and an extremely low initial setting for resource depletion.

Whilst the model is a simplified representation of a real system, with a number of underlying assumptions, it is unlikely that such an outcome would have been predicted without running the model.

Take Home Message

The message is very clear. Greater equality in our society may well be essential to our survival. At a social level, Wilkinson and Pickett in their book the “Spirit Level” show very clearly that the more equal societies in the developed world do far better on every social measure than those societies with an unequal distribution of wealth. See Wilkinson TED lecture