Solution to Finding equilibria in 2-D systems (by Aiqing - Final Review Session)

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Nov 24, 2024

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Finding equilibria in 2-D systems 1. Consider a two population system modeling inter-species competition between frogs (X) and salamanders (Y) with change equations: X’ = 5X - X 2 - XY Y’ = 10Y - Y 2 - 6XY Identify the equilibrium points of the system. Step 1: Calculate the X-nullclines and Y-nullclines X-nullclines: X’ = X(5 - X - Y) = 0 X = 0 or Y = 5 - X Y-nullclines: Y’ = Y (10 - Y - 6X) = 0 Y = 0 or Y = 10 - 6X Step 2: Make a table to check their intersections X-nullclines X = 0 Y = 5 - X Y-nullclines Y = 0 (0, 0) Y = 5 - X Y = 0 -> X = 5 (5, 0) Y = 10 - 6X X = 0 Y = 10 - 6X -> Y = 10 (0, 10) Y = 5 - X Y = 10 - 6X -> X = 1, Y = 4 (1, 4) Thus, the equilibrium points are . (? * , ? * ) = (0, 0), (5, 0), (0, 10), (1, 4)

2. Step 1: Calculate the M-nullclines and S-nullclines M-nullclines: M’ = M(3 - M - 2S) = 0 M = 0 or M = 3 - 2S S-nullclines: S’ = S (4 - 2M - 2S) = 0 S = 0 or S = 2 - M Step 2: Make a table to check their intersections M-nullclines M = 0 M = 3 - 2S S-nullclines S = 0 (0, 0) M = 3 - 2S S = 0 -> M = 3 (3, 0) S = 2 - M M = 0 S = 2 - M -> S = 2 (0, 2) M = 3 - 2S S = 2 - M -> M = 1, S = 1 (1, 1) Thus, the equilibrium points are . (? * , 𝑆 * ) = (0, 0), (3, 0), (0, 2), (1, 1)

3. Step 1: Calculate the L-nullclines and F-nullclines L-nullclines: L’ = L(- F + 4) = 0 L = 0 or F = 4 F-nullclines: F’ = F (5L - L 2 - F) = 0 F = 0 or F = 5L - L 2 Step 2: Make a table to check their intersections L-nullclines L = 0 F = 4 F-nullclines F = 0 (0, 0) F = 4 F = 0 impossible F = 5L - L 2 L = 0 F = 5L - L 2 -> F = 0 (0, 0) F = 4 F = 5L - L 2 -> L = 1 or L = 4 (1, 4), (4,4) Thus, the equilibrium points are . (? * , 𝐹 * ) = (0, 0), (1, 4), (4, 4)

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Solution to Finding equilibria in 2-D systems (by Aiqing - Final Review Session)

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