In general, populations are considered to be regulated if they do not grow without bounds to infinity (i.e., not N → ∞). This crude definition will not be satisfactory to many ecologists. Stochastic fluctuations in birth and death rates may prevent a population from ever achieving infinite size, but that does not mean that stochastic fluctuations can regulate a population’s size. There’s a big difference between affect and regulate. Stochastic fluctuations in birth and death rates will certainly affect overall population growth rate, but they will not contribute to regulating the population. Regulation in a general sense implies control of population size: in other words, a predictable change in population growth rate as population size changes. A slightly more specific definition would be that a population’s total growth rate decreases as population size increases. This definition should also not be very satisfactory though for the same reasons. Population growth rate may decline with increasing population size, but if population growth rate never becomes negative the population will continue to grow without bounds, albeit with an ever slowing rate of increase. Population regulation typically implies that at sufficiently high population abundances total population growth rate becomes negative. If total population growth rate is negative, population size will decline because this means that the per capita birth rate is lower than the per capita death rate. Population regulation may also imply that at some range of population sizes, typically small population size, total population growth rate is positive. This would tend to maintain the population. A population cannot go extinct if it will increase when it becomes small.

Features of a model of population regulation

In our analyses of population growth models that are density independent, we considered models with no population regulation. Density independence implies no population regulation mechanisms are operating. How can we alter these models to build in one or more mechanisms of population regulation? First, let’s consider the general features of a population with regulation. As we decided above, regulation implies that total population growth rate is positive when N is small (so that the population will increase when small), and total population growth rate is negative when N is very large (so population size will decrease). The figure here represents this situation. If this is true, the total population growth rate function must become zero at some intermediate value between these two areas (assuming that it is a continuous function).   dN/dt versus N The value of N where {dN \over dt} = 0 has a special name. This point is called an equilbrium.

An equilibrium is a population size where total population growth rate is zero. (Note that if total population growth rate is zero, then per capita population growth rate must also be zero.) We will symbolize a value of population size that is an equilibrium by a superscript asterisk – N* in this case.

A single model may have more than one equilibria, and even models that have no population regulation can have equilibria. For example, in the model of exponential growth (i.e., {dN \over dt} = r N ) are there any equilibria, and if so, what are they? Equiibria are identified by setting the total population growth rate equation equal to zero and solving for all values of N that satisfy this equation. For {dN \over dt} = r N = 0 , the only value of N that satisfies this equation is N=0. Thus, the existence of equilibria are not indicative of population regulation. The equilibrium illustated in the last figure is a stable equilibrium. Stable equilibria are indicative of population regulation.

A stable equilibrium has the following property – if the population is started anywhere close to this equilibrium, population size will approach the equilibrium.

In the above figure, imagine that population size is at a value that is just below the equilibrium (i.e., N < N*). In this vicinity, total population growth rate is positive, so population size increases (i.e., population size will move towards the equilbrium). Likewise, now imagine that the population size is started at a value just above the equilibrium (i.e., N > N*). In this vicinity, total population growth rate is negative, so population size decreases (i.e., population size will move towards the equilibrium). Thus, everywhere this population could be started in the vicinity of the equilibrium, the population will move to the equilibrium – thus, this equilibrium is stable.

Also, consider how total population size is changing with population size at this stable equilibrium. If we imagined taking the derivative of this function with respect to N at N*, is the derivative positive or negative? It is negative. (Remember that the derivative of a function at a point is the slope of the tangent line to the function at that point.) This is the mathematical definition of a stable equilibrium.

{\partial{dN \over dt} \over \partial N} < 0

An equilibrium can also be unstable. The figure here illustrates an unstable equilibrium. At population sizes below N*, total population growth rate is negative, so population size declines. At populatoin sizes above N*, total population growth rate is positive, so population size increases. Thus, population size will move away from the equilibrium on both sides. dN/dt versus N unstable In this case, {\partial {dN \over dt} \over \partial N} > 0 . This is the mathematical definition of an unstable equilibrium.

Stable equilibria tend to attract the population size, and unstable equilibria tend to repel the population size.

A General Model of Population Regulation

Although total population growth rate is used to identify the locations of equilibria and to assess whether they are stable or unstable, the critical model features that are needed for population regulation are per capita demographic rates that change with population size.

A demographic rate is density dependent if its value changes with population size. Thus, birth rate is density-dependent if the per capita birth rate in a population changes with population size.

Thus, population regulation implies density dependent per capita demographic rates. This figure shows four differnet relationships between per capita birth and death rate in one population. Density dependent birth and death rates giving stable equilibrium The upper left panel illustrates a population in which both birth and death rates are density-independent. Note that neither changes with population size. This population would increase exponentially at a constant rate. Why? Because birth rate is greater than death rate at all population sizes. In the other three panels, one or both of these demographic rates are density dependent. In the upper right panel, per capita death rate is constant, but the per capita birth rate decreases with increasing population size. Where these two rates are equal (i.e., b = d) the population has an equilibrium. Is this equilibrium stable? Yes, because b > d below the equilibrium, and b < d above the equilibrium. What do these relationships in per capita rates mean for total population growth rates in these two areas? You should be able to show that this equilibrium is stable based on {\partial{dN \over dt} \over \partial N} < 0 . The lower two panels show similar scenarios. In the lower left panel, per capita birth rate is constant with population size, and per capita death rate increases with population size. In the lower right panel, both per capita birth and death rates are density-dependent. In both of these panels, the resulting equilibria where the rates are equal (i.e., where the lines cross) are stable. Can you prove this? Use the same reasoning as in the last paragraph. In each of these panels, one other equilibrium exists. Here’s a hint: it is the same value of N in each panel. Where is it? It is N=0. With no immigration, N=0 is always an equilibrium. Now consider the next figure of similar panels. Density dependent birth and death rates giving unstable equilibrium In the upper left panel in this figure, will the population increase or decrease from any N? It will decline to N*=0 from all N>0. This makes N*=0 a stable equilibrium! In all the other panels in this figure, the equilibria with N*>0 are all unstable equilibria as well. Why? Explain why each is an unstable equilibrium based on the derivative of total population size with respect to N at each, and based on the relationships of per capita birth and death rates above and below the equilibrium point. The existence of density-dependent birth and death rates do not guarantee any kind of population regulation. Can you think of a way to draw both the per capita birth and death rates to be density dependent, but still have an overall density-independent rate of population growth. (Hint: remember that r = bd, and you want to draw a figure where r remains constant but b and d change with population size.)

Convert these graphs into a mathematical model

Let’s now build a general population growth model that can handle any of the situations illustrated in the last two figures. To do this, we must first specify functions for the per capita birth and death rates. Since these were lines in every panel, we will define simple functions of lines for each.

b = b_{0} + \beta N

and

d = d_{0} + \delta N

All four parameters defining the slopes and intercepts of these functions can take positive or negative values. Thus, the per capita birth and death rate functions in all eight panels in the last two figures can be described by these two simple functions. We will use the simple continuous time model of population growth as substitute our functions for per capita birth and death rates into it to derive our model

{dN \over dt} = r N = (b - d) N

= ((b_{0} + \beta N) - (d_{0} + \delta N)) N

= (b_{0} - d_{0} + \beta N - \delta N) N

= (b_{0} - d_{0}) N + (\beta N - \delta N) N

= (b_{0} - d_{0}) N + (\beta - \delta) N^{2}

Now that we have a general model, let’s analyze it. As you will see, the same approach is taken to the basic analysis of every population model, whether only one species or a set of interacting species is being modeled. Thus, we will walk slowly through the process here in repeatable steps.

1. Find all equilibria.

The first step in model analysis is to find all the equilibria. To do this set the total population growth rate equal to zero and find all N ≥ 0 (i.e., populations cannot have negative abundances) that satisfy this equation. Obviously, N*=0 is an equilibrium, since substituting N=0 into the population growth rate equation gives dN/dt=0. To determine whether other equilibria exist, we need to do some algebra. Thus, the following algebra ensues

{dN \over dt} = (b_{0} - d_{0}) N + (\beta - \delta) N^2 = 0

(b_{0} - d_{0}) N + (\beta - \delta) N^2 = 0

(b_{0} - d_{0}) + (\beta - \delta) N = 0

(\beta - \delta) N = -(b_{0} - d_{0})

N^* = -{(b_{0} - d_{0}) \over (\beta - \delta)} = {(d_{0} - b_{0}) \over (\beta - \delta)} = {(b_{0} - d_{0}) \over (\delta - \beta)}

The equation for the equilibrium population size shows a very interesting relationship between the slopes and intercepts of the per capita birth and death rate functions. For the equilibrium to exist, the relationships between their slopes and intercepts must be reversed: the function with the more positive slope must also have the lower intercept. If this is not true, the function above will give N*<0, which is not allowed, and so the only equilibrium will be N*=0 in that case. Since these are lines, the maximum number of times they can intersect is once. Thus, we have at most two equilibria for this model: N*=0 and N*=(b0-d0)/(δ-β).

Before we move on, think a bit about what it would mean if the only equilibrium present were N*=0. What does that imply about the graphs of per capita birth and death rates above? Answer Either they are parallel, or they cross at a value less than zero.

2. Evaluate the stability of each equilibrium.

To evaluate the stability of each equilibrium, we must evaluate the value of the derivative of the total population growth rate with respect to population size at each equilibrium (see the above figures and the logic we used to evaluate them). For our model here, this derivative evaluated at an equilibrium is

{\partial {dN \over dt} \over \partial N} = {\partial ((b_{0} - d_{0}) N - (\beta - \delta) N^{2}) \over \partial N}

= b_{0} - d_{0} + 2 (\beta - \delta) N| _{N = N^*}

= b_{0} - d_{0} + 2 (\beta - \delta) N^*

To test each equilibrium, we simply plug in its value into this equation, and determine whether the derivative is greater than or less than zero. Remember that if {\partial{dN \over dt} \over \partial N} < 0 the equilibrium is stable, and if {\partial{dN \over dt} \over \partial N} > 0 the equilibrium is unstable. First, consider the equilibrium that always exists at N*=0. Substituting N*=0 into this equation shows that the stability of the equilibrium at N*=0 depends only on the relationship between the intercepts of the per capita birth and death functions, which makes perfect sense. If the birth rate function has a higher intercept than the death rate function, N*=0 is unstable because b0d0>0, which makes {\partial{dN \over dt} \over \partial N} > 0 . In contrast, if the death rate function has a higher intercept than the birth rate function, N*=0 is a stable equilibrium because b0d0<0, which makes {\partial{dN \over dt} \over \partial N} < 0 . Think about what these two results mean for a minute. If per capita birth rate is greater than per capita death rate (i.e., b0d0>0), the population will increase. In this case, if the population is very small (i.e., N≈0), the population will move away from the equilibrium, which is the definition of an unstable equilibrium. This is why N*=0 is an unstable equilibrium if b0d0>0. Conversely, if per capita birth rate is lower than per capita death rate (i.e., b0d0<0), the population will decrease. Thus if this is true and the population is very small (i.e., N≈0), the population will move to the equilibrium, which is the definition of a stable equilibrium. The practical sense of evaluating the stability of an equilibrium is as follows:

If you start a population anywhere near an equilibrium and the population will move to the equilibrium, the equilibrium is stable. Otherwise, if you can identify at least one location near an equilibrium where the population will move away from the equilibrium, the equilibrium is unstable.

Now consider the equilibrium at N*>0. Substitute from above the value of N* into the partial derivative of dN/dt with respect to N, and the following algebra ensures:

{\partial {dN \over dt} \over \partial N} = b_{0} - d_{0} + 2 (\beta - \delta) N^*

= b_{0} - d_{0} + 2 (\beta - \delta) (- { b_{0} - d_{0} \over \beta - \delta})

= b_{0} - d_{0} - 2 (b_{0} - d_{0})

= - (b_{0} - d_{0})

Remember that this equilibrium is stable if this function is negative at this point. This function is negative at the equilibrium if the intercept of the per capita birth rate function is greater than the intercept of the per capita death rate function: i.e., b0d0>0. Think about all the things this means. First, this function describes how the total population growth rate changes as N changes. If this function is negative at this point, it means that the tangent line at this point has a negative slope. Second, since this is an equilibrium, the function dN/dt=0 at this point. Putting these two facts together, this means that dN/dt must be greater than zero at values of N just below N*, and dN/dt must be less than zero at values of N just above N*. This is the only way one can mathematically reconcile these two facts. This is exactly the conditions described in the first figure above. In contrast, this equilibrium is unstable if the function is positive at this point. This function is positive at this point if b0d0<0, meaning that is has a positive tangent slope at this point. From the same train of logic, the function will look like the second figure above. (As an exercise for yourself, return to all the possible ways you could draw the per capita birth and death rate functions, and see if this mathematical analysis gives you exactly the same answer as when you evaluate the graphs directly.)

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