## Stochastic Models

Deterministic models of population growth allow the population size at time t + 1 to be calculated based on the knowledge of Nt plus the other factors that are included in the models. Stochastic models attempt to include the probabilistic nature of biological systems by representing one (or more than one) variable in the given model as a random variable. Natural populations can start at the same size but due to chance increase or decrease at different rates based on the value of the random variable at each time step. Any variable in a model that has any degree of uncertainty can be parametrized as a random variable. Stochastic variables can be based on a probability of occurrence (i.e., probability of breeding at any given time step, or probability of having a certain number of offspring during a given reproductive bout). These probabilities are generally assigned based on previous knowledge of the system. Stochastic models where both births and deaths occur randomly allow for a chance of extinction. Figure 6 depicts two simulations of a random birth-death model (eqn [7]) where the probability of births and deaths occurring during a given time step was 0.5 (probabilities of occurrence were independent of each other). This representation exemplifies the nature of stochastic population models for small population sizes.

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Figure 6 Two simulations of a hypothetical population using a stochastic birth-death model. The probability of birth or death occurring at each time step was 0.5. The rest of the parameters were: N0 = 1.2, b = 1.2, d = 0.8.

The first simulation went extinct after 10 time steps while the second simulation did not go extinct during the simulation. Small populations are at a greater risk of extinction due to their inability to rebound after a catastrophic event. Stochastic models are often used to estimate the probability of extinction for a given population. This probability is based on several repetitions of the same stochastic model.

Random variables can also be chosen from a frequency distribution of historical data or from a statistical distribution fitted to historical data - Figure 7 was created using the latter case, depicting growth curves for a hypothetical population created with both deterministic and random models. The same parameters were used except for the random model the birth rate was chosen at the beginning of each time step from a normal distribution with a mean of 1.2 and standard deviation of 0.1 while the birth rate for

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Figure 7 Population growth for a deterministic model (heavy line) and model with the birth rate was a random variable (dotted line) chosen from a normal distribution with a mean of 1.2 and a standard deviation of 0.1.

the deterministic model was 1.2. The population size of the stochastic model oscillates around K but does not converge on a single equilibrium point.

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