Appendix A. Local stability analysis of the general model equation (Eq. 11).
We introduce the functions
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(A.1)
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with partial derivatives,
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(A.2a)
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(A.2b)
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(A.2c)
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(A.2d)
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Specifying the partial derivatives to the extinction equilibrium gives the characteristic equation . This gives the condition for population extinction in this system
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(A.3)
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We now shift our attention to the non-trivial equilibrium. Using Eq. 12 and the condition in Eq. 11 that either , we can write the Jacobian for this system at the non-trivial equilibrium
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(A.4)
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where . This gives the characteristic equation
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(A.5)
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In general, the non-trivial equilibrium will be stable when
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(A.6)
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Note that the discriminant is a quadratic in c with a positive coefficient for the quadratic term. Therefore setting the discriminant to zero and solving for c gives us the values between which the discriminant is negative,
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(A.7)
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when it is, condition (A.5) becomes
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(A.8)
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When the discriminant is positive, we distinguish two cases. In the first case, the quantity is positive and condition (A.5) simplifies to
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(A.9)
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which is always true. In the second case, the quantity is negative and condition (A.6) becomes
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(A.10)
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If the quantity on the r.h.s. of (A.10) is negative the condition is never true. Therefore, if
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(A.11)
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the equilibrium is unstable. If the r.h.s. of (A.10) is positive then the condition for stability becomes
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(A.12)
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If then this is always true. If then the equilibrium is stable when
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(A.13)
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This concludes the local stability analysis of the two equilibria.