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A Menagerie of Population Dynamics

General Introduction

The overall dynamics exhibited by a population depend upon an number of factors. The types of dynamics we will consider here are ones that are determined by density dependent feedback within a population and are purely determistic. As a population increases in size it exherts a negative influence on reprodutive output and survivorship, acting to lower population growth rates. The tendency is to push the population towards its carrying capacity. However, if density dependent feedback is very strong and the potential reproductive output in the absence of density effects is high, a wide variety of dynamics can be observed. The types of dynamics observed will depend, in fact, upon the relative strengths of reproductive output and density effects. As the impact of density intensifies or reproductive capacity increases, the dynamics exhibited by a population displaced away from its carrying capacity will shift from a smooth approach to the carrying capacity, known as monotonic damping, to more complex dynamics, such as damped oscillations, limit cycles, and eventually to the most extreme cases of density dependence, which exihibit a pattern known as chaotic dynamics.

In the examples presented here, we will look at the dynamics of a very simple population model, shown above, as they depend upon the magnitude of parameter values. The model is based upon a standard form of a yield-density equation, appropriate for modeling an annual plant population. The effects of density as depicted by the model are intensified with increases in the parameter b above values of 1.0. Maximum reproductive output in the absence of density effects are represented by the parameter R. The parameter a does not affect the dynamics and is used to express density on the appropriate scale.

Monotonic Damping

A population that is damped monotonically will exhibit a smooth approach to its carrying capacity. This will happen if the population is either above or below its carrying capacity. This is equivalent to the classic case of logistic growth discussed in most traditional ecology textbooks.

An Example Model

Damped Oscillations

A population that is characterized by damped oscillations will overshoot and then undershoot the carrying capacity in sequence. The degree to which it undershoots or overshoots, however, will lessen with each timestep, until the population eventually settles on the carrying capacity.

An Example Model

Limit Cycles

A population that is characerized by a limit cycle will alternate between 2, 4, or possibly 8 population sizes in succession. In the example shown here the population is alternating between two population sizes. The larger population size is above the hypothetical carrying capacity of the population. Density effects are strong. resulting in a population far below carrying capacity during the next timestep. Under these circumstances density effects are low and the population size increases to above the carrying capacity once again. The population continues to overshoot and undershoot the carrying capacity in succession.

An Example Model

Chaos

A population will exhibit chaotic behavior, if reproductive output is high and there are strong density effects regulating population size (i.e., large values for the exponent in the divisor of our example model). The apparent behavior of the model will appear to be extremely erratic, however, the underlying process is purely deterministic.

An Example Model

The Relationship between Observed Population Dynamics and the Parameters R and b in our Simple Population Model

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