![predator vs prey game predator vs prey game](https://www.gamespace.com/wp-content/uploads/2021/09/Aliens-vs-Predators-1.jpg)
I used numerical integration and placed the results into Javascript typed arrays. Reducing the predator population slightly can either increase or decrease the maximum predator population, but reducing it by a large amount almost always increases the amount of predators! Is there a way to know when the population will decrease vs increase? At the time I wrote this page, I hadn’t gotten to that part of the book yet, but the answer is yes: page 60, Modeling Life 1.5. See what happens to the predators and also the prey (blue). Try it near the bottom and the top of the cycle. Try this: click/drag in the middle and set the predator (red) population. I wanted to see for myself, so I wrote the simulation: Starting prey: □ predators: □ The textbook says that it depends on which phase of the cycle you’re in when you reduce the population. Reducing the predator population can cause the predator population to go up higher than it had been.
![predator vs prey game predator vs prey game](https://pbs.twimg.com/media/Ec_uWZDU8AI_lqD.jpg)
One of the more fascinating things I read in the Modeling Life textbook is that there’s counterintuitive behavior when intervening in the population: page 7, Modeling Life, with my highlights+annotation Similarly, I can calculate the min and max numerically but I don’t see an analytical way to calculate this. I don’t see this mentioned on the Wikipedia page or the textbook.īut I discovered by calculating and plotting the mean that it’s equal to the equilibrium! The mean should be the average value over one cycle. There’s also an equilibrium point when both predator and prey are at 0, but that’s not so interesting. This both means the model can be surprising (good for math!), and that the model likely does not match reality (bad for biology!). The Wikipedia page says the period is \(2π / \sqrt α\), but it doesn’t! And the equilibrium for predators does not depend on the death rate of predators. The period of the cycle is how long it takes before it repeats. I did read a little bit about how to draw vector ( this paper and this blog post) and I experimented a little bit. It’d be a cool animation but it’s a bit of work so I didn’t make it. But viewed from the end, it’s a phase plot. Viewed from the side or top, it’s a plot of predators or prey over time. Both of these 2d diagrams are views of a 3d diagram (time, predator, prey), which might be fun to make but I didn’t get around to doing so. This shows how the two variables related, but not how fast they’re moving. There’s a different type of chart, the phase diagram, which plots the two populations together but without time: The chart above shows the populations over time. The rules give us these differential equations which could be implemented in code: It also has some surprising behavior that I want to explore in the simulations. This model is widely studied and seems like a good starting point for me to learn this type of system. These rules aren’t a particularly good model but they’re simple.
![predator vs prey game predator vs prey game](https://i.ytimg.com/vi/YXj039q8FkA/maxresdefault.jpg)
To go deeper into the math I’ve been watching Robert Ghrist’s lectures. During the covid pandemic I’ve been learning about dynamic systems in biology from a textbook, Modeling Life.