Lotka-Volterra Model


Lotka-Volterra Model

The mass action approach to modeling trophic interactions was pioneered, independently, by the American physical chemist Lotka (1925) and Italian mathematician Volterra (1926). These authors argued that consumer and resource populations could be treated like particles interacting in a homogeneously mixed gas or liquid and, under these conditions, the rate of encounter between consumers and resources (the reaction rate) would be proportional to the product of their masses (the "law of mass action"). The dynamics of the interaction between a resource population H and a consumer population C is then described by the differential equation

where a represents the per-capita rate of change of the herbivore in the absence of consumers, b the consumption rate of the consumer, c the constant of conversion of resources into consumer offspring, and m the per-capita mortality rate of consumers in the absence of resources. The terms in the equations have the following meaning:

  1. aH = the growth rate of the herbivore population in the absence of predators. Thus, in the absence of predators, the herbivore population grows according to the equation dH/dt = aH, which integrates to Ht = H0eat , the exponential growth equation.
  2. bHC = the rate of consumption of herbivores, or their death rate due to attack by predators.
  3. cbHC = the rate of production of predator offspring, which is directly related to the number of prey consumed.
  4. mC = the death rate of consumers in the absence of food. Thus, in the absence of prey, predators die according to dC/dt = -mC, which integrates to Ct = C0e-mt , an exponential decay equation.

Because the exponential growth law applies to all populations all of the time it is usually more interesting and useful to study the per-capita rate of change of each population (it is also more biological meaningful because this variable describes how individual organisms respond to their environments). We do this by dividing each equation by the numbers of individuals in the population

where Rh is the per-capita rate of change of the prey and Rc is that for the predator. These equations are now known as the R-functions for the prey and predator populations, respectively.


Population Dynamics

The general nature of the dynamics resulting from the interaction between two species can be deduced by calculating the zero-growth isoclines for the two species. It is obvious that prey and predator populations will be in equilibrium when their per-capita rates of change are zero because births will equal deaths under this condition. Hence, we can find the equilibrium isoclines, the lines where each species is constant, by solving the dynamic predator-prey equations when Rh = Rc = 0. Starting with the L-V prey R-function (above) we have

In other words, the prey zero-growth isocline is a constant that intercepts the predator axis at a / b.

Likewise the predator zero-growth isocline

is also a constant that intercepts the prey axis at m / cb. These isoclines are shown in the figure below.

 

Figure. (Left) Herbivore (H) and carnivore (C) phase space showing zero-growth isoclines for the L-V model (herbivore isocline = horizontal blue line, carnivore isocline = vertical red line). Arrows show directional vectors of population change in the four regions of phase space. (Right) Herbivore (blue) and carnivore (red) time series plot showing cycles of abundance with the predator cycles lagging behind the prey.

 

Notice that a community equilibrium occurs where the two isoclines intersect because, at this point, both populations remain unchanged. Predator-prey phase space is divided into four regions by the equilibrium isoclines:

  1. In this region both populations grow and the vectors of population change point towards the upper right corner.
  2. In this region the predator population grows while the prey declines and the vectors of population change point towards the upper-left corner.
  3. In this region both populations decline and vectors point towards the lower-left corner.
  4. In this region the prey population grows while the predator declines so that all vectors point to the lower-right.

Notice that the vectors cross the prey isocline vertically and the predator isocline horizantally, because variables do not change when they are on their own isoclines. Also note that a dynamic trajectory starting at any point in phase-space will form a closed orbit whose amplitude is determined by the starting point. This is known as a neutrally stable limit cycle. If we plot the densities of prey and predators against time we obtain cycles of abundance of both species with the predator cycle lagging behind that of its prey. Similar cycles are often observed in the field and are sometimes used in support of L-V theory.


References

Lotka, A. J. 1925. Elements of physical biology. Williams and Wilkins, Baltimore.

Volterra, V. 1926. Pages 409-448 in Chapman R. N. 1931. Animal ecology. McGraw-Hill, New York.


©1998 Alan A. Berryman