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(a) First suppose that the two types have the same intrinsic growth rate r = r₁ = r2 but that they differ in their sensitivity to crowding: dny dt dn₂ dt = rm1 rn₂ (₁ n₁ + n₂ K₁ n] + n₂ K₂ where K₂ > K₁. Draw the nullclines and direction field for this system on the ni, n2 state space. On your diagram, identify the equilibrium point where any non-trivial starting point (n₁, n₂ # 0) will ultimately end up

(a) First suppose that the two types have the same intrinsic growth rate r = r₁ = r2 but that they differ in their sensitivity to crowding: dny dt dn₂ dt = rm1 rn₂ (₁ n₁ + n₂ K₁ n] + n₂ K₂ where K₂ > K₁. Draw the nullclines and direction field for this system on the ni, n2 state space. On your diagram, identify the equilibrium point where any non-trivial starting point (n₁, n₂ # 0) will ultimately end up

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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(a) First suppose that the two types have the same intrinsic growth rate r = r₁ = r2 but that they differ in their sensitivity to crowding: dny dt dn₂ dt = rm1 dny rn₂ (₁ dt dn₂ dt n₁ + n₂ K₁ where K₂ > K₁. Draw the nullclines and direction field for this system on the ni, n₂ state space. On your diagram, identify the equilibrium point where any non-trivial starting point (n₁, n2 # 0) will ultimately end up n₁ + n₂ K₂ (b) Now set K₁ = K₂ = K but suppose that each type is less sensitive to crowding from the other type than from their own type (e.g. this could be because the two types don't use exactly the same resources) - = rm₁ (1-1₁+0₁ 2 (1- K a2n1+n₂) K Draw the nullclines and direction field for this system for the case where a₁ = a2 = 1 and identify the equilibrium point where any non-trivial starting point (n₁, n2 # 0) will ultimately end up.
Transcribed Image Text:(a) First suppose that the two types have the same intrinsic growth rate r = r₁ = r2 but that they differ in their sensitivity to crowding: dny dt dn₂ dt = rm1 dny rn₂ (₁ dt dn₂ dt n₁ + n₂ K₁ where K₂ > K₁. Draw the nullclines and direction field for this system on the ni, n₂ state space. On your diagram, identify the equilibrium point where any non-trivial starting point (n₁, n2 # 0) will ultimately end up n₁ + n₂ K₂ (b) Now set K₁ = K₂ = K but suppose that each type is less sensitive to crowding from the other type than from their own type (e.g. this could be because the two types don't use exactly the same resources) - = rm₁ (1-1₁+0₁ 2 (1- K a2n1+n₂) K Draw the nullclines and direction field for this system for the case where a₁ = a2 = 1 and identify the equilibrium point where any non-trivial starting point (n₁, n2 # 0) will ultimately end up.
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