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Each system of differential equation 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 describe 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.)(a) $ \frac {dx}{dt} = 0.12x - 0.0006x^2 + 0.00001xy $$ \frac {dy}{dt} = 0.08x + 0.00004xy $(b) $ \frac {dx}{dt} = 0.15x - 0.0002x^2 - 0.0006xy $$ \frac {dy}{dt} = 0.2y - 0.00008y^2 - 0.0002xy $
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Calculus 2 / BC
Chapter 9
Differential Equations
Section 6
Predator-Prey Systems
Missouri State University
Oregon State University
University of Michigan - Ann Arbor
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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Each system of differentia…
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Competition and cooperatio…
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The system of differential…
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The following system of di…
Okay, so this program is two parts on for each part. We're going to determine the relationship within two species. For part a notice that eso Let's talk about the general principle for the x d y on the x d t n d y t t this system of differential equation to determine the relationship between the two x suspicious action. Why we only look at the crossing term so which is a term contains X and y. So we look at the coefficients of X and y. So for the first equation, we have C one X y, And for the second equation, we have C two x y and plus some other terms but not crossing term, and C one C two for convenience. We define there are both positive because we have both positive relationship between the crossing term. That means thes two species or cooperative because the presence of one species is helping the presence of the other species and the for part B. Um, So again, we use the same principle here The X over d t in close to that, you say three x way, plus the other term regarding ex Mm I can say this Apple Macs and the Y O. D. T equals two. They can see four x y pull us in G y so we can see that for positive 33 and C four. The coefficients here are both negative, which means the presence of one species is hurting or at some impacting the that of the other species, which means these two species are competitive.
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