Logistic Models
I. Classic Logistic Models After realizing limitations of the exponential model, we may wonder: can we do better? We know that the resources are not unlimited. Therefore, at some point, the growth rate must slow down and eventually the population should reach the place where the environment cannot hold more. Based on the above description, it is reasonable to come up to a graph like this:
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Figure 1 An explanation of logistic growth model Adapted from Lumen Boundless Biology 2013
Mathematically, we have a model, which is called Logistic equation.
Figure 2 An explanation about the growth rate changes over time Adapted from Lumen Boundless Biology
For the moment, let’s just ignore the equation f(x) on the right. We’ll cover the mathematical part in the later section. Right now, we’re focusing on the actual meaning of this graph. Let’s check together whether this graph has all the features we discussed before.
By taking the tangent line of each pair of points along the x-axles, we find that the slope is indeed getting smaller and smaller. And when x is sufficiently large, the slope is close to 0. It means, the growth rate of population decreases in each period of years. And after a long time, the number of populations becomes stable. It makes sense intuitively. Please be aware that the decline of growth rate does not indicate the drop of population. On the contrary, the total population is still increasing as shown in the graph. For more advanced readers, the growth rate is represented by the derivative of the function. And the carrying capacity is the limit of the function when x approaches infinity.
The actual formula was first proposed by the Belgium mathematician Pierre-François Verhulst in 1835, who stated “The virtual increase of population is limited by the size and the fertility of the country. As a result, the population gets closer and closer to a steady state” in his Note on the law of population growth. (1838 as cited in Bacaër, 2011,)
Here N, K, r represent population, carrying capacity, and a growth rate parameter respectively. Noted the formula is given in the differentiation form.
This model assumes that the growth rate r is linear and it is decreasing in terms of N. In other words, r_a=F(N). For a more complete model, we will introduce r_m as the max growth rate. But it is beyond our text slope.
Interestingly, if N(t) is much smaller than K, this ordinary equation is approximately as our previous exponential model. Here comes another question. Can we write N(t) explicitly just like the exponential one?
The exciting answer is yes! But understanding how to get it requires some math background in an introductory level ordinary differential equation class. We strongly encourage interested readers take MA366 or MA 266 held by Purdue math department.
Let’s do some math!
Figure 3 Merma, 2004 P80 The logistic equation Adapted from Northwestern University website
And we’ll see in later section that even the complicated logistic equation cannot predict the population accurately.
II.Theta Logistic model
III.Logistic Model with Allee’s Effect
