![]() Below, Figure 1 shows the differences in the natural (exponential) growth rate versus the logistic growth rate of the rabbit population.įigure 1: the “Exponential” curve shows the exponential growth of rabbits using the form, while the logistic model shows the classic “S” shape that levels off at the “Carrying Capacity” and has the form. Without this term, the rabbit population would just grow exponentially since the constant represents the natural growth rate of the rabbit population. ![]() Notice that as approaches, this logistic functions tends to zero. Knowing this, we can look at the value of and it should be a large, positive integer as it is a number that denotes the maximum limit of the rabbit population that can be sustained in the given environment. Here, we have a logistic function that, in ecological population growth modelling, represents the change in the rabbit population with respect to the carrying capacity of the environment in which our rabbit and fox populations are changing. However, this is being multiplied by another term, Since is a constant, we have, as we saw in the Exponential Growth post, exponential growth. On the right hand side (RHS) of the equation, we see that the population,, is being multiplied by the ratio. Equation (1) gives the change in the rabbit population over time, Secondly, we can look at the equations (1) and (2) and make some inferences from what it is doing. It isn’t necessary to know what all these parameters or variables represent to solve this problem–save that they’re initial conditions–but we can try and parse the given information to perhaps venture a good extrapolation.įirstly, in both our DEs we should realize that the terms and are in fact functions of their respective populations with respect to time,, such that and. Then, we are given five parameters that are, for the most part, ratios except for. We’re given two coupled DEs, one that represents the change in the rabbit population,, and the other that represents the change in the fox population. It is clear that much of the work here is done for us in terms of setting up of the problem, e.g. Generate a computer plot of the rabbit and fox populations as a function of time using the following parameters: Here, is the rabbit population as a function of time, and is the fox population as a function of time. The problem that follows is a typical homework question for a differential equation and/or mathematical physics course (for more information on this exact problem a source for it can be found here and the full textbook here–this text was found while writing this blog and does a really good job explaining the problem).* Problem: Foxes & RabbitsĪ model for a predator-prey system (Rabbits and Foxes in this case) utilizes the following two differential equations: ![]() This type of problem usually involves population ecology models using two or more coupled DEs commonly referred to as Lotka-Volterra, or in our case, Competitive Lotka-Volterra equations. In our case, we will look at the infamous Foxes and Rabbits scenario. However, in that example we were only dealing with a single changing quantity over time (the population of bacteria) but, what if we have more than one quantity changing and each changing quantity is dependent on the other changing quantity? This time we’ll delve into a popular problem and further explore the wizardry of differential equations.Ī useful example is the common predator-prey model, where the changing population of the predator is proportional to the changing population of the prey over time and vice-versa. We’ve talked about differential equations (DEs) previously here and worked out a problem involving exponential growth of a hypothetical bacterial colony.
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