10.4. Predator-Prey Model with Functional and Numerical Responses

   Now we are ready to build a full model of predator-prey system that includes both
   the functional and numerical responses.

   We will start with the prey population. Predation rate is simulated using the
   Holling's "disc equation" of functional response:

                                      [eq16.gif]

   The rate of prey consumption by all predators per unit time equals to

                                      [eq17.gif]

   The equation of prey population dynamics is:

                                      [eq18.gif]

   Here we assumed that without predators, prey population density increases
   according to logistic model.

   Predator dynamics is represented by a logistic model with carrying capacity
   proportional to the number of prey:

                                      [eq19.gif]

   This equation represents the numerical response of predator population to prey
   density.

   The model was built using an Excel spreadsheet:

   [1][icoexcel.gif] Excel spreadsheet "predfunc.xls"

   You can modify parameters of this model to simulate various patterns of
   population dynamics. Differential equations are solved numerically, and it may
   happen that the algorithm (2nd order Runge-Kutta method) will not work for some
   combination of parameter values. Thus, change parameters with caution. If you
   suspect that the algorithm does not work properly, reduce the time step (cell A5)
   until results become independent from the time step.

   Simulation results are presented below. This model exhibits more various dynamic
   regimes than the Lotka-Volterra model.

   [gdynam2.gif] r[H] = 0.2
   K = 500
   a = 0.001
   Th = 0.5
   r[P] = 0.1
   k = 0.2
   No oscillations
   [gdynam1.gif] r[H] = 0.2
   K = 500
   a = 0.1
   Th = 0.5
   r[P] = 0.1
   k = 0.2
   Damping
   oscillations
   converging to
   a stable
   equilibrium
   [gdynam.gif] r[H] = 0.2
   K = 500
   a = 0.3
   Th = 0.5
   r[P] = 0.1
   k = 0.2
   Limit cycle

   This model can be used to simulate biological control. The goal of biological
   control is to suppress the density of the pest population using natural enemies.
   We will assume that the prey in our model is a dangerous pest, and that the
   predator was introduced to suppress its density. Withour predators the density of
   prey population is equal to the carrying capacity, K = 500. After a predator with
   a search rate, a = 0.001, was introduced, the equilibrium population density,
   N^*, declined to the value of 351. Beddington et al. (1978, Nature, 273: 573-579)
   suggested to measure the degree of pest suppression by the ratio:

                                    [eqbedd.gif] .

   For example, if a = 0.001, then q = N^*/K = 351/500 = 0.7. Biological control is
   successful if the value of q is low (at least,