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191

CHAPTER 2

Linearity and Nonlinearity

SECTION 2.6

Systems of DEs: A First Look

191

In this system the equilibria on the axes are all unstable, so the populations always head toward a ⎡ 200 ⎤ coexistence equilibrium at ⎢ ⎥ . See Figure, where x and y are measured in hundreds. ⎣300 ⎦

„

Finding the Model

Example of appropriate models are as follows, with real positive coefficients. 13.

x′ = ax − bx 2 − dxy − fx y′ = −cy + dxy

14.

x′ = ax + bxy y′ = cy − dxy + eyz z ′ = fz − gx 2 − hyz

15.

x′ = ax − bx 2 − cxy − dxz y′ = ey − fy2 + gxy z′ = −hz + kxz

„ 16.

Host-Parasite Models (a) A suggested model is H 1+ P P′ = −bP + dHP

H ′ = aH − c

where a, b, c, and d are positive parameters. Here a species of beetle (parasite) depends on a certain species of tree (host) for survival. Note that if the beetle were so effective as to wipe out the entire population of trees, then it would die out itself, which is reflected in our model (note the differential equation in P). On the other hand, in the absence of the beetle, the host tree may or may not die out depending on the size of the parameters a and c. We would probably pick a > c , so the host population would increase in the absence of the parasite. Note too that model says that when the parasite (P) population gets large, it

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689  

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689  

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...

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