The Allee Effect

Posted: August 26th, 2021

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The Allee Effect

Introduction/Summary

The topic, “The Allee Effect,” is the main concept that is described, with a significant focus on mathematical equations utilized, understanding the behaviors of a population. The Allee Effect refers to a biological process that is expressed by assessing the correspondence between population size or density and the average individual fitness. The behavior is mostly associated with populations that are sparsely distributed. The concept is based on dependence on population density that is comparatively low. In this case, a strong Allee Effect occurs in situations when the population sinks to low densities and negative. Conversely, weak Allee Effect exhibits increasing and positive proliferation rate. However, decreasing rates of proliferation with lower but positive density is a null hypothesis. Therefore, the two forms of Allee Effect differs based on the way each exhibit critical population density or size within a particular region.

Further, populations characterized by strong Allee Effect often show critical population size or density as the population grows negative. On the contrary, when a population exhibits weak Allee Effect it has a decreasing per capita growth rate which is at low rate. Equally, the same population will show positive per capita rate of growth. Thus, these conclusions imply that once a population hits a threshold, it will become extinct except when there are other human or environmental mechanisms to affect the trend.

Questions 1 – 3

Question 1 (a)

Therefore;

Which isthe logistic equation, has the following solution;

Where;

And, however, for since r =1, then

Graphically;

Figure 1: sigmoid growth curves

As sigmoid has high r-value, it is seen from the graph, and high r-value makes an easy jump in the value of N (t), sigmoid growth recovers quickest.

Question 1 (b)

The logistic equation will be given as follows;

Whereby,  shows the population growth rate,  is the maximum per capita growth rate for a population while  is the individual in a population and  shows that are carrying capacity. The graph for this case will be as follows;

Figure 2: Sigmoid growth curve

As such, different sets of r yield different curves as exhibited in the following figure;

Figure 3: Sigmoid growth curves for Sharks and Sardines

However, according to the question, there are two types of species, that is, Sardine with high r-value and Sharks with lower r-value. Therefore, according to the graphs above, it can be clearly explained that species having a higher r-value have a faster rate of recovery compared to those with lower r values. In this case, Sardine is going to recover faster than Sharks.

Question 2

Given

The equation is of the form;  and the equilibria are the N such that f (N) = 0. Therefore, f (N) = 0 and →N=0, N =0

 => N =K

 => N = A. Therefore, the three values where the population is at equilibrium are 0, K, and A.

Question 3

The logistic equation is as below;

. Whereby A is referred to as the Allee threshold. Further, N (t) =A shows the size of the population below which growth in the community becomes negative as a result   Allee Effect. This is located at the N value between N=0 and N=K, expressed as  based on the species type. Therefore;

Whereby the equilibrium points are N=0, N=K. Hence, the phase portrait can be given as;

Figure 4: Equilibria curves

From the figure, it can be shown that the equilibrium point N=0 and N=K are stable points.