Competitive Species With a Parameter

We will be performing some matrix calculations. I am using the Linear Algebra, rather than the linalg, package.

with(LinearAlgebra):

Differential equations.

xdot := x*(2-x-a*y); ydot := y*(3-y-2*a*x);

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Find equilibria. Note that three of the equilibria are independent of a, but the fourth equilibrium is in the first quadrant if and only if a < 2/3 or 3/4 < a.

solve({xdot=0, ydot=0}, {x, y});

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fsolve(-2+3*a=0,a); fsolve(-3+4*a=0,a); fsolve(-1+2*a^2=0,a);

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NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtbkdGJTY5UTUmdW1pbnVzMDsuNzA3MTA2NzgxMkYoLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRigvJSVzaXplR1EjMTJGKC8lJWJvbGRHUSZmYWxzZUYoLyUnaXRhbGljR0Y4LyUqdW5kZXJsaW5lR0Y4LyUqc3Vic2NyaXB0R0Y4LyUsc3VwZXJzY3JpcHRHRjgvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRigvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYoLyUnb3BhcXVlR0Y4LyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R1EldHJ1ZUYoLyUpY29tcG9zZWRHRjgvJSpjb252ZXJ0ZWRHRjgvJStpbXNlbGVjdGVkR0Y4LyUscGxhY2Vob2xkZXJHRjgvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGKC8lKm1hdGhjb2xvckdGQy8lL21hdGhiYWNrZ3JvdW5kR0ZGLyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRigvJSltYXRoc2l6ZUdGNS1JI21vR0YlNjNRIixGKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRk0vJSdsc3BhY2VHUSQwZW1GKC8lJ3JzcGFjZUdRM3Zlcnl0aGlja21hdGhzcGFjZUYoLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGWC8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZDLyUrYmFja2dyb3VuZEdGRi1GLTY5US0wLjcwNzEwNjc4MTJGKEYwRjNGNkY5RjtGPUY/RkFGREZHRklGS0ZORlBGUkZURlZGWUZlbkZnbkZpbkZcbzcjNiQkISs3eTFycSEjNSQiKzd5MXJxRl5y

Find Jacobian.

J := <<diff(xdot,x) | diff(xdot,y)>, <diff(ydot,x) | diff(ydot,y)>>;

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Neither Survive Equilibrium. Source for all a > 0.

J0 := subs({y = 0, x = 0}, J); DD := Determinant(J0); T := Trace(J0); T^2-4*DD; Eigenvectors(J0);

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Only Species Y Survives Equilibrium. Saddle if a < 2/3; sink if 2/3 < a; never a spiral sink although at a = 5/3 the sink has only one straight-line solution..

J1 := subs({x = 0, y = 3}, J); DD := Determinant(J1); T := Trace(J1); disc := simplify(T^2-4*DD); E1 := Eigenvectors(J1);

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solve(DD=0); solve(disc = 0);

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Only Species X Survives Equilibrium. Saddle if a < 3/4; sink if 3/4 < a; never a spiral sink although a sink with a single straight line solution at a = 5/4..

J2 := subs({y = 0, x = 2}, J); DD := Determinant(J2); T := Trace(J2); disc := simplify(T^2-4*DD); E2 := Eigenvectors(J2);

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solve(DD=0); solve(T=0); solve(disc = 0);

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Both Species Survive Equilibrium. Both coordinates are positive if and only if a < 2/3 (sink) or 3/4 < a (saddle).

eq := {x = (-2+3*a)/(-1+2*a^2), y = (-3+4*a)/(-1+2*a^2)}: J3 := subs(eq, J); DD := Determinant(J3); T := simplify(Trace(J3)); disc := simplify(T^2-4*DD); E3 := simplify([Eigenvectors(J3)]):

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

solve(DD=0); solve(T=0);

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

plot(disc,a=0..2/3); plot(disc,a=3/4..1);

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

NiUtSSdDVVJWRVNHNiI2JDdTNyQkIjMrKysrKysrK3YhIz0kIiIlIiIhNyQkIjMzbm1UJilHXGF2RiskIjM2ejgmUSgpSGtKJSEjPDckJCIzVExla2BvIT5nKEYrJCIzI1x6czpYMydbWEY0NyQkIjNDbTt6PilHX2woRiskIjMpcC1KJFwmZlB4JUY0NyQkIjM9bTthUVUhKjN4RiskIjMnKSpvRnFYbjIoXEY0NyQkIjNhTGVSWlhLaXhGKyQiM3AneWhebERROSZGNDckJCIzem0iej4sXz0ieUYrJCIzMyxzITRtIip5RyZGNDckJCIzWypcN0ckWzhqeUYrJCIzTTsmbygpSFFOVSZGNDckJCIzL24ieiUqZnJoInpGKyQiM1VWSjAlXGs9YiZGNDckJCIzMytEY3cjUSFwekYrJCIzPy1kJCpRJCpwcGNGNDckJCIza0xMM18iPU0tKUYrJCIzSTY9R3BQNCN5JkY0NyQkIjNTbTsvd2ZKciEpRiskIjM7SEdtNkR5dWVGNDckJCIzJioqKipcN2VQXzcpRiskIjMncCpbI1xeYkkoZkY0NyQkIjNzKioqXFBmIVF6IilGKyQiMzdxL2RuZTdtZ0Y0NyQkIjNlKioqXCg9dWJKIylGKyQiMzs8UClwdic+XmhGNDckJCIzUW0ielcoKlEqeSMpRiskIjN2K25BVUwqXEEnRjQ3JCQiM3NMTDNGLUdOJClGKyQiM1QkW0MlejAnKjNqRjQ3JCQiMylITEwkZSczSVEpRiskIjNJd1ZDMVJBeGpGNDckJCIzTitESnEiRyZRJSlGKyQiMz9XW0Z2Om9ga0Y0NyQkIjNpTEwkZU1zd1spRiskIjM0OSRcKm8jeSo9bEY0NyQkIjNoK0RKJkgiZlQmKUYrJCIzaUxdLCM0LiQpZSdGNDckJCIzJykqXFBmKVskSGYpRiskIjNBVV1VQlhGX21GNDckJCIzeUwkZWtgMWxrKUYrJCIzLVplYVNTNDxuRjQ3JCQiM2VMZSpbLi1kcClGKyQiM01VYDxJUS52bkY0NyQkIjNrbTsvRWd3WygpRiskIjM1Tj4oUU5yZiRvRjQ3JCQiMyJmO3olKmYlKVEhKSlGKyQiM19MLV0mPSZwKCpvRjQ3JCQiM2cqXChvemEnPSYpKUYrJCIzRUxfVUFFQV1wRjQ3JCQiMzRuO3pXaG8uKilGKyQiM0p4SSFSUip5MHFGNDckJCIzTyoqKipcaT5BZCopRiskIjNtcEMjPng6PzEoRjQ3JCQiMz4qKlxpOmpmNCEqRiskIjMqUS1yczVXZjYoRjQ3JCQiM0UqXDdgPnItMSpGKyQiMy1VTTgxaDxuckY0NyQkIjN5KipcUDRxYDsiKkYrJCIzJSp5VC4qPVVJQShGNDckJCIzO0xMJGVNJTRuIipGKyQiM3E+K0ViVFFzc0Y0NyQkIjNuKioqXFA0djVBKkYrJCIzWiJwSyxOR1VLKEY0NyQkIjNjbSJ6V24qKSpwIypGKyQiMy9mIUdacCwwUChGNDckJCIzVCoqKlw3Qm1NSypGKyQiM0RYT1QoKW9PP3VGNDckJCIzIUckZWsuTnl0JCpGKyQiMyFlbzFTbUttWShGNDckJCIzUytEY14mempVKkYrJCIzKHo5QlhFX1ZeKEY0NyQkIjNDTEwzLT0heVoqRiskIjNbS1A3LF1TZ3ZGNDckJCIzKikqXDdHOE87YCpGKyQiM2RbTDZEQCwzd0Y0NyQkIjNXbW1tIipcWyRlKkYrJCIzbUcrOkZkSWB3RjQ3JCQiM0ttO2FRel1PJypGKyQiMydbaVxBYiYzKnAoRjQ3JCQiM3VLZWtHPTQqbypGKyQiM3kmSHYsJD0oUnUoRjQ3JCQiM1EqKioqXGk0VFAoKkYrJCIzJSpHLHNCX3kleShGNDckJCIzL0wkM0Y5IXojeipGKyQiMykqPihwQSVSMkp5RjQ3JCQiM0ltbW1UPktVKSpGKyQiM1UxQSp5MV4/KHlGNDckJCIzMStESnFKOCYqKSpGKyQiM0hNOioqUXdKOnpGNDckJCIzMSt2b2Etb1gqKkYrJCIzcSJ5YShbXUxjekY0NyQkIiIiRi4kIiIpRi4tSSdDT0xPVVJHRiU2JkkkUkdCR0YlJCIjNSEiIiRGLkYuRmFbbC1JK0FYRVNMQUJFTFNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JFEiYUYlUSFGJS1JJVZJRVdHRmRbbDYkOyQiKysrKyt2ISM1RmZ6SShERUZBVUxUR0Yl

plot(T,a=0..2/3); plot(T,a=3/4..1);

subs(a=2/3,eq);

subs(a=3/4,eq);