Nicholson-Bailey Model

The Nicholson-Bailey model was developed in the 1930's to describe the population dynamics of a coupled host-parasite (or predator-prey) system. The model uses difference equations to describe the population growth of both populations. The model assumes that parasites search for hosts at random, and that both parasites and hosts are assumed to be distributed in a non-contagious ("clumped") fashion in the environment. As originally described, the model does not allow for stable host-parasite interactions. To add stability, the model has been extensively modified to add new elements of host and parasite biology. The model is closely related to the Lotka-Volterra host-parasite model, which uses differential equations to describe stable host-parasite dynamics. Here, we will review the derivation of the model and examine two modifications to the model. Please refer to the reading list for more detailed information on the model and its modifications.

Model Derivation

As originally described, the model takes the form:

         N_t+1 = LN_t e^-aP_t                                              [1.]

         P_t+1 = N_t [1-e^-ap_t]                                           [2.]

where:

         N_t = number of hosts (prey) at time, t

         P_t = number of parasites (predator) at time, t

         a = area of discovery

         L = host reproductive rate

         e = exponential constant.

To derive the model, we first define the number of encounters between host and parasite N_e as:

         N_e = a N_t P_t                                                   [3.]

The parameter "a" was termed the "area of discovery", which was thought to be a species-specific constant defining the lifetime rate of encounters between hosts and a single searching parasite. Mathematically, we can define "a" as:

         a = N_e/N_t                                                       [4.]

To distribute encounters among hosts, Nicholson-Bailey assumed that parasites searched at random, and thus would re-encounter hosts previously attacked -- or, for predators, encounter space previously searched. They used a Poisson model to distribute encounters (N_e) among hosts (N_t). The general form of the Poisson model is:

         P(X) = e^-A_x ( A_x ^X / X!)                                  [5.]

where:

          P(X) = probability of X occurrences

          A_x= average occurrence of X

          X! = X factorial

Solving for "zero" occurrences e.g., probability of not being attacked, gives:

         P(0) = e^-A_x  ( A_x ⁰ / 0!)                                 [6.]

            = e^-A_x

Setting A_x = N_e/N_t, gives:

         P(0) = e^-(N_e/N_t)                                              [7.]

Since N_e/N_t is defined as the "area of discovery" (Eq. 4), multiplying by the number of searching parasites, P_t, gives the probability of a host escaping attack by P_t parasites (i.e., being encountered "zero" times):

         P(0) = e^-aP_t                                                   [8.]

and the probability of being attacked (i.e., being encountered one or more times) is:

         P (>0) = 1- e^-aP_t                                            [9.]

Inserting these statements of probability into equations of host and parasite population growth gives the Nicholson-Bailey model:

         N_t+1 = LN_t e^-aP_t

         P_t+1 = N_t [1-e^-aP_t]

Model Behavior

Two questions of interest concerning host-parasite models are: (a) does the model reach an equilibrium, and (b) is that equilibrium stable? These are important questions for host-parasite models, as it is thought that, in Nature, host parasite systems are stable over long time periods, as evidenced from studies of natural control. For the Nicholson-Bailey model, to determine its equilibrium point, one sets: N_t+1 = N_t and P_t+1 = P_t (i.e., there is no change in numbers of hosts and parasites from time t to time t+1). Solving the equations in this fashions shows that the model does reach an equilibrium point defined by N* and P*. However, this equilibrium is not stable, as any perturbation (e.g., addition of parasites or hosts), drives one or both populations to extinction (see Fig. 1). Later modifications to the model attempted to bring it stability, by adding new elements of host and parasite biology. Here we provide two examples of these efforts: (a) mutual interference and, (b) the functional response. This effort to modify the Nicholson-Bailey model to make it stable has been criticized (e.g. Murdoch et al. 1985).

Figure 1.