This is because more prey makes it easier for predators to find prey and also because an increase in predators makes it harder for the prey to hide. The rate is proportional to both x and y. The second term, -βxy, is negative and thus describes death. The α term is a constant that controls how fast the exponential growth occurs. The growth rate is proportional to x, so we know it is positive exponential growth (as seen in previous lessons). The first term, αx, is positive and therefore defines growth. The top equation defines how the population of prey, x, changes in relation to the predator and prey populations. Given that both x and y are functions of time, we have the following system of ODEs: Because both predator and prey start with ‘p’, we will use the variable x to refer to the prey population and the variable y to refer to the number of predators. This gives us two populations each with two separate terms to add to our equation. Predators also die off over time due to age. Predators are dependent on prey for sustenance and thus grow at a rate dependent on both the predator and prey population. The difference is that prey are also killed off by the predators at a rate directly proportional to both the predator and prey population. Prey multiply exponentially, similar to our exponential example in the previous lessons. There are two populations in question: the predators and the prey. We’ll start with a simple Lotka-Volterra predator/prey two-body simulation. Today, we’ll put that knowledge to good use. Previous posts explained how numerical solutions work and how Matlab will perform the calculations for you automatically. Finally, the series will conclude with a post on model fitting and a post about chaotic systems. Today we’ll look at two simulations of living systems (Lotka-Volterra and SIR). Regardless, I’m finally back in the swing of things and ready to write up Part 3! To recap: Lesson 1 and Lesson 2 looked at how ODEs are solved numerically and how higher order solutions are more accurate than naive implementations. It’s been a crazy summer that has included some vacation time plus an overseas trip to a conference. Well, I feel like I should apologize for such a long delay between posts.
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