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For each predator-prey system, determine which of the variables, $ x $ or $ y, $ represents the prey population and which represents the predator population. Is the growth of the prey restricted just by the predators or by other factors as well? Do the predators feed only on the prey or do they have additional food sources? Explain.(a) $ \frac {dx}{dt} = -0.05x + 0.0001xy $$ \frac {dy}{dt} = 0.1y - 0.005xy $(b) $ \frac {dx}{dt} = 0.2x - 0.0002x^2 - 0.006xy $$ \frac {dy}{dt} = -0.015y + 0.00008xy $

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(A).The predator population increases only through the term $0.0001 x y$, that is, by encounters with the prey and not through additional food sources.(B). The predator population increases only through the term $0.00008 x y$, that is, by encounters with the prey and not through additional food sources.

Calculus 2 / BC

Chapter 9

Differential Equations

Section 6

Predator-Prey Systems

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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For each predator-prey sys…

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The system of differential…

So in this problem, um, we look at our formulas for predator prey equations, and we recognize that C one X Y is positive. So that means that the existence of this is going to be good for species X, whereas C two X Y is positive as well. So that means that the existence is beneficial for the species. Why eso using that? We want to see what kind of model this is. It could be cooperative, competitive or predator prey. Eso Based on the system that we have, we see that see one is positive and C two is negative. Tells us that, um, why is helping X But X is hurting. Why? So that means that X is creditor. And why is Cray Um so with that, in the absence of prey, the predators will decrease. Um, and there's no other. That means that there's no other food source for the predator, then moving on to the next part. What we see is that when the population increases for why, uh, X will decrease and an increase in X will increase. Why? So this tells us once again it's a predator prey relationship. We also want to notice that, um, as long as X isn't too large, um, it will be okay, But if X gets extremely large, then it will start causing issues within the X population. So that tells us that small increases in X will result, um, in a lower growth rate over time on bats, a za result of overcrowding. I'm not enough. Resource is in the prey population. But then also, we see that when there's an absence of the predators, the prey will experience more logistic growth instead of exponential growth. And that's because eventually, um, there's so much space containing them that they'll run out of resource is

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