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Imagine a large number of habitat patches, N, of which X(t) are occupied at time t. The remainder, N - X(t), are empty and available for colonization by individuals dispersing from the occupied patches. Imagine that each occupied patch produces potential colonizers at the rate c. These propagules settle, randomly, on any patch; a fraction (N-X)/N land on empty patches that they colonize. The total colonization rate is then cX(N-X)/N. The differential equation for the number of occupied patches is N X X' = CX x (^~^) - ₁ (² - dX. N (a) Explain the effect of the term -dX, assuming that d is positive. Give a plausible physical inter- pretation of the constant d.

Imagine a large number of habitat patches, N, of which X(t) are occupied at time t. The remainder, N - X(t), are empty and available for colonization by individuals dispersing from the occupied patches. Imagine that each occupied patch produces potential colonizers at the rate c. These propagules settle, randomly, on any patch; a fraction (N-X)/N land on empty patches that they colonize. The total colonization rate is then cX(N-X)/N. The differential equation for the number of occupied patches is N X X' = CX x (^~^) - ₁ (² - dX. N (a) Explain the effect of the term -dX, assuming that d is positive. Give a plausible physical inter- pretation of the constant d.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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here is the b question:

Find the value of X after one year using Euler’s method to solve the differential equation for the following parameters: c = 3.4 year−1, d = 1.6 year−1, N = 1000, and X(0) = 10, using a time step of h = 0.5 years.

However, do question C only 

= 800. (c) (Advanced) The figure below plots the numerical solution to the differential equation for the parameters in (b). On the same figure, sketch what the solution would look like if X(0) Explain the long time behaviour of this solution to the differential equation. 1000 800 600 400 2 3 4 Time t (years) Number of occupied patches X 200 1 LO 5
Transcribed Image Text:= 800. (c) (Advanced) The figure below plots the numerical solution to the differential equation for the parameters in (b). On the same figure, sketch what the solution would look like if X(0) Explain the long time behaviour of this solution to the differential equation. 1000 800 600 400 2 3 4 Time t (years) Number of occupied patches X 200 1 LO 5
A paper by Mullineaux et al. [3] discusses models for meta-populations in the region around deep sea hydrothermal vents. These create a “patchwork" of sea floor habitats that host their own ecosystems. Quoting from Ref. [3]: Imagine a large number of habitat patches, N, of which X(t) are occupied at time t. The remainder, N- X(t), are empty and available for colonization by individuals dispersing from the occupied patches. Imagine that each occupied patch produces potential colonizers at the rate c. These propagules settle, randomly, on any patch; a fraction (N-X)/N land on empty patches that they colonize. The total colonization rate is then cX(N − X)/N. The differential equation for the number of occupied patches is N X X' = cX ex (~ ~ *) - - dX. N (a) Explain the effect of the term −dX, assuming that d is positive. Give a plausible physical inter- pretation of the constant d.
Transcribed Image Text:A paper by Mullineaux et al. [3] discusses models for meta-populations in the region around deep sea hydrothermal vents. These create a “patchwork" of sea floor habitats that host their own ecosystems. Quoting from Ref. [3]: Imagine a large number of habitat patches, N, of which X(t) are occupied at time t. The remainder, N- X(t), are empty and available for colonization by individuals dispersing from the occupied patches. Imagine that each occupied patch produces potential colonizers at the rate c. These propagules settle, randomly, on any patch; a fraction (N-X)/N land on empty patches that they colonize. The total colonization rate is then cX(N − X)/N. The differential equation for the number of occupied patches is N X X' = cX ex (~ ~ *) - - dX. N (a) Explain the effect of the term −dX, assuming that d is positive. Give a plausible physical inter- pretation of the constant d.
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