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Competition and cooperation Each system of differential equations is a model for two species that either compete for the same resources or cooperate for mutual benefit (flowering plants and insect pollinators, for instance). Decide whether each system describes competition or cooperation and explain why it is a reasonable model. (Ask yourself what effect an increase in one species has on the growth rate of the other.)$$\begin{aligned} \text { (a) } \frac{d x}{d t} &=0.12 x-0.0006 x^{2}+0.00001 x y \\ \frac{d y}{d t} &=0.08 x+0.00004 x y \\ \text { (b) } \frac{d x}{d t} &=0.15 x-0.0002 x^{2}-0.0006 x y \\ \frac{d y}{d t} &=0.2 y-0.00008 y^{2}-0.0002 x y \end{aligned}$$

(a) The system represents cooperation.(b) The system represents competition.

Calculus 2 / BC

Chapter 7

Differential Equations

Section 5

Systems of Differential Equations

Campbell University

University of Nottingham

Idaho State University

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in this problem. We have two species x and y and party in part B. Each describes a different system corresponding to the dynamics between the two species X and Y, and we were asked to classify the models in parts A and B according to whether they describe cooperation between the two species or competition between the two species. Now what we can understand from looking at the terms of the model is that the X Y terms that is, a product terms between the two species represent encounters between them. So whenever they interact, we're going to have a term like extends. Why, poppet? And this comes from something called the law of Mass Action, which is implicit in the formulation of the ordinary differential equations for the model. And we can consider how these encounters describe competition cooperation no and increase in why he's going to make DX over DT larger for party increasing. Why is gonna make the ex already t larger because of the positive term. 0.1 actually 0.1 x y. So whenever X and y interact, we have that the D X over DT growth rate is incremental by this much. So that means the ex grocery is gonna be higher because of this interaction. And an increase in X, conversely, makes d y over dt larger because of the positive term, 0.0 four x Y. So this means that either species impacts the growth rate of the other species and a positive cooperative way. So the system describes a cooperation model as a whole. Now, for part B, we see the opposite because an increase in X part B reduces the growth rate of why so it causes D Y over dt to go down because of the negative. Come on, negative interaction term. So you have a negative coefficient on X and wire, not negative, because their populations of species. So this is automatically negative and similarly, an increase in why reduces the growth rate of X so d x over DT equals don't because of this term negative 0.0 006 x Y So this system describes a competition model because the growth rate of one species is in Worsley, proportional to, um, the increase in the other species or, more precisely, said we have that one species increasing causes the growth rate of the other species to go down

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