Goals: explore ways in which the logistic growth model can be modified to model specific population conditions.  Consider how time lags can cause population cycles.  Evaluate how intraspecific competition can result in non-linear decline in per capita growth rate with population size, how to model non-linear decline, and how it impacts the rate of population growth.

The Lecture: Logistic growth is ageneral model in that the basic pattern fits many species fairly well.  For particular species, it can be modified to make it more accurate.  We will consider two ways of doing this in today's lecture.

First, we will consider time lags. A time lag occurs when the population growth rate does not respond immediately to current environmental conditions; it responds to conditions that exist in the past.  For example, maybe the birthrate in a population depends on the population size when individuals got pregnant, and there is a gestation period so that the growth that depends on this occurs at some time later.  The kind of lag we will focus on is in resourse response to population size.  For various reasons, response of a resource, such as plant growth for herbivores, may not respond to the population size immediately.  Perhaps the population of herbivores one year determines how much plants will grow in the following year, for example.  Food may be plentiful in the current year but if the plants are damaged and do not grow back, it will be scarce in the following year.

To model this kind of time lag, we can add the lag, represented by the Greek letter tau (t), to our model.  t indicates the time it takes before the population growth rate responds to the population size.  Thus, the density dependent part of the logistic growth equation will be related to the population size tau time units before the present.  N_t-t represents the population size t time units before the present.  The formula for logistic growth with a time lag is:

dN/dt = r_mN_t(K-N_t-t)/K

The result of a time lag is that a population will grow above the carrying capacity, since it will not respond immediately to being at the carrying capacity.  Tau time units after it reaches the carrying capacity, growth will be zero: the population will respond to being at K.  After that, growth will be negative since the population will start responding to being above K.  It will then decline below K, since it will not respond immediately to being at K again.  Thus, time lags lead to oscillations of populations around the carrying capacity.

There are three kinds of oscillation that can occur.  Damped oscillations occur if the population initially cycles around K, but cycles become smaller with time and eventually the population stays at K.  This is shown here:

Stable limit cycles occur when the population continues to oscilate around K.  This is shown here:

Unstable oscillations occur if the population cycles become larger and larger.  These result in extinction.  The most common pattern is to have just one cycle: the population grows well above K and then crashes to zero.  This is shown here:

Damped oscillations represent a fairly stable situation; stable limit cycles are less so (populations go through small sizes during which they may be vulnerable to extinction), and unstable oscillations are the least stable (since they lead to extinction.)  Let's consider what affects the stability of oscillations.  In general, the farther above the carrying capacity that a population grows, the less stable the situation will be, since a population far above K will deplete resources and this will lead to decline far below K.  Two factors from the model affect how far above K the population grows:

• increase in the length of the time lag, tau, results in populations growing for a longer period before they respond to being above K.  As a result, increases in tau will cause populations to grow far above K and will decrease stability (make larger oscillations leading to stable limit cycles or perhaps unstable cycling and extinction.)
• larger r_m  values occur if populations reproduce at a higher rate; higher reproduction means that within a given time a population will grow larger.  So increase r_m values result in populations growing farther above K and will decrease stability (make larger oscillations leading to stable limit cycles or perhaps unstable cycling and extinction.)

As a second modification to logistic growth, remember that when we developed the model of logistic growth we assumed that the decrease in per capita growth rate with population size was linear.  This means that the impact of a new individual on decreasing birthrate and increasing death rate of any other individual is the same in small and large populations.  There are likely to be situations in which this is not true.  Per capita population growth could be relatively unaffected by population size for small populations and be more strongly affected by increased populations close to the carrying capacity.  For example, suppose the limiting resource is nest or den sites.  It might be that there is little impact of population size on per capita population growth until these are close to being all taken; at that point, there would be a very strong impact on per capita population growth.  In other words, intraspecific competition decreases the per capita growth rate disproportionately strongly at large populations that are approaching the carrying capacity.

We can model this non-linearity of response using the following equation:

dN/dt = r_mN(1-(N/K)^q).  The exponent is the Greek letter theta.  Theta does not have a specific ecological meaning the way the time lag, tau, does.  The effect of theta is to allow us to model a situation in which per capita growth depends on population size in a way that is non-linear.

Note first that if theta=1, this equation is the same as the regular logistic growth equation, so theta=1 gives a linear decline in per capita growth rate with population size.  If theta >1, then the per capita growth rate only declines slightly at small population sizes, and declines much more rapidly as the population reaches K.  This is the model that gives the situation described above, in which intraspecific competition decreases the per capita growth rate disproportionately strongly at large populations that are approaching the carrying capacity.  It is also possible to give a value of theta between 0 and 1.  This would model a situation in which per capita growth is more strongly affected by more individuals in smaller populations than larger populations.  This seems unlikely to apply to many situations, although there are a few peculiar species where it might apply.  For example, there are ant species which change their development so that when populations are small individuals produced are very large (for ants) but when populations grow large, tiny individuals are produced.  Otherwise, there do not seem to be many situations in which it would be reasonable to include a theta value less than 1.

The following figure shows the decline in per capita growth with population size for theta less than 1, equal to one, and greater than 1.

Theta values also affect how rapidly a population reaches carrying capacity, as shown here:

Finally, theta values affect the population size at which population growth is maximum.  This is shown here:

Note that for theta>1, the peak in population growth occurs at a population size of greater than K/2.  This would be important for situations (such as those described in the previous lecture) when we are interested in managing populations so that they are maintained at the size that gives maximum population growth.