In the logistic model, population growth is affected by density factors, as noted by the term (Nt/K) in the logistic equation (Equation 1 below).
Nt+1 = Nt + r Nt (1 – Nt/K) (1)
In this equation, K is the carrying capacity, r is the per-individual population growth rate under ideal conditions, Nt is the population size at time (t), and Nt+1 is the population size at the next time period (t+1). The effects of density are felt immediately, and the population grows according to the standard logistic equation: close to exponentially when density is low (N < K), and slowing down as N approaches K, the carrying capacity.
As we saw in Chapter 10 of the textbook, the effects of density can be delayed, and this delay can affect population dynamics. We can model the delayed effect of density by modifying the logistic equation as in Equation 2 below.
Nt+1 = Nt + r Nt (1– N(t–d)/K) (2)
The parameter (d) is a measure of the delay of the density effects. If d is two days, then the effects of density experienced during the third day would become apparent in the fifth day.
Whether and how the delayed density dependence affects population dynamics depends on the product of two factors: the per-individual growth rate of populations under optimal conditions (r), and the time delay (d). At low values of rd, the population still experiences logistic growth. At somewhat higher values of rd, the population will overshoot the carrying capacity, and will then fall below the carrying capacity. It will continue to oscillate, but each oscillation will be closer to the carrying capacity. These are called dampened oscillations. At still higher values of rd, the oscillations will not be dampened. Instead, the population will cycle between high and low densities at regular intervals.
In the simulation exercises, we will see how changes in both the growth rate and the time delay alter the population dynamics. Here, we will use a strain of the hypothetical species of beetles, Tribolium wadeii, whose per-individual growth rate is 0.15/day. Suppose that by altering food supply to different life stages that we can manipulate the time delay of the effects of density-dependence.
Open the simulation in a new window
Question 1
What happens when there is no time delay by setting d = 0 and running the simulation. What results do you obtain? What type of growth is exhibited? What, if any, is the carrying capacity?
Question 2
What happens when we allow for a two-day time delay of the density effects? Set d to equal 2 and start the simulation. What results do you obtain? What type of growth is exhibited?
Question 3
What happens when the time delay is 4 days? Set d to equal 4 and start the simulation. What results do you obtain? What type of growth is exhibited?
Question 4
What happens under time delays of 8 and 12 days?
Question 5
Another strain of T. wadeii has a population growth rate of 0.2. Repeat the simulations with delays of 2, 4, and 8 days. What results do you obtain?
Question 6
Advanced exercise: Investigate where the transition point between logistic growth and dampened oscillations occurs by adjusting values of r and d. Do the same for dampened oscillations and a stable limit cycle.
