Let $a,b,e$ be fractions of patches occupied by the two competitors and empty, with $a+b+e=1$. Assume well-mixed propagule dispersal, equal independent fire extinction rate $m>0$, colonization constants $c_A>c_B>0$, and net replacement advantage $q=k_B-k_A>0$ for the poorer colonizer. The Levins metapopulation model with replacement isReplacement fluxes $k_{B}ab$ and $k_{A}ab$ cancel in the total occupancy equation. At every face the outward flux is zero or points into the patch probability simplex, so these equations preserve valid fractions. The constants include both propagule arrival and establishment probability; the model does not require unequal fire susceptibility.

Put $\Delta c=c_A-c_B$. Solving the two occupied-patch balances and their sum constraint givesThus strict coexistence is equivalent toToo little disturbance lets the better competitor exclude the colonizer; too much disturbance favors colonization or destroys occupancy. A positive equilibrium automatically has $m<c_A$. If the poorer colonizer cannot survive alone, coexistence can nevertheless be possible because it repeatedly replaces established patches of the better colonizer.

These bounds also have an invasion interpretation. The $A$-only equilibrium is $a=1-m/c_A$ when $c_A>m$; the growth rate of rare $B$ there is $q-m(q+\Delta c)/c_A$, positive at the upper-bound condition above. At a feasible $B$-only equilibrium the rare-$A$ growth is $-q+m(q+\Delta c)/c_B$, giving the lower bound. At coexistence the occupied-patch Jacobian matrix has trace $-c_A{a}_{\ast }-c_B{b}_{\ast }<0$ and determinant $a_{\ast }{b}_{\ast }q(q+\Delta c)>0$, proving local asymptotic stability. The Bendixson-Dulac criterion with multiplier $1/(ab)$ gives divergence $-c_A/b-c_B/a<0$ in the interior, excluding periodic competition cycles.

To specify selection, a monotone tradeoff alone is insufficient: one must say how two different trait values determine replacement. A natural model uses a trait $x$ with increasing competitive ability $B(x)$ and decreasing colonization $c(x)$, and net replacement $B(y)-B(x)$ by a mutant $y$ against a resident $x$. The resident metapopulation occupies $1-m/c(x)$, requiring $c(x)>m$. The rare-mutant per-capita growth, or invasion fitness, isIt is zero at $y=x$. Differentiating with respect to the mutant trait gives the selective pressureHigher colonization is valuable in empty habitat; higher competitive ability is valuable where the resident already occupies patches. This is colonization-competition tradeoff invasion fitness.

At an interior candidate for a local evolutionarily stable strategy, $g(x_{\ast })=0$. A strict, nondegenerate local evolutionarily stable strategy follows ifsince nearby distinct mutants then have negative invasion fitness. For a strict global evolutionarily stable strategy, every feasible distinct mutant must satisfyThis is the supporting-line condition on the tradeoff curve. At a trait boundary replace the vanishing-gradient condition by the appropriate one-sided inequality. Equality against another mutant, or zero local curvature, needs the second condition of evolutionary stability or higher-order terms; the strict tests above are sufficient and are the usual nondegenerate conditions, not a claim that every tie is unstable.

For evolutionary approach to a singular point, the selection gradient must also point toward it, locally $g^{\prime }(x_{\ast })<0$. This is logically distinct from testing mutant fitness. In this particular additive-ability model with $c^{\prime }<0<B^{\prime }$, negative mutant curvature implies negative $g^{\prime }$ becauseIf replacement instead has a general kernel $\kappa (y,x)$, replace $B(y)-B(x)$ in $f$ by that kernel and differentiate it; monotonicity does not justify importing the displayed additive formula without that modeling assumption. The tradeoff shape, disturbance rate and replacement law together determine selection and local or global uninvadability.

The basic reproduction number is the expected value of secondary infections caused by one typical infectious individual introduced into an otherwise susceptible population. For a heterogeneous model it is the dominant eigenvalue of the next-generation operator, so the appropriate infected-type distribution is part of “typical.” A small introduction grows when ${\mathcal{R}}_0>1$ and decays when ${\mathcal{R}}_0<1$, under the usual linear invasion assumptions.

The asymptotic susceptible level depends on the model. In a homogeneous SIS model with transmission rate $\beta$ and recovery rate $\gamma$, the positive endemic equilibrium has $S_{\ast }=1/{\mathcal{R}}_0$ when ${\mathcal{R}}_0=\beta /\gamma >1$. In a closed SIR model, however, the final size relation for an epidemic isFor an infinitesimal seed in a wholly susceptible population this becomes $-\log S_{\infty }={\mathcal{R}}_0(1-S_{\infty })$. The epidemic peak occurs at $S=1/{\mathcal{R}}_0$, and a nontrivial outbreak leaves $S_{\infty }<1/{\mathcal{R}}_0$. Thus $1/{\mathcal{R}}_0$ is an endemic or instantaneous invasion threshold, not a universal final susceptible fraction.

For cooperative infections, let $s,x,y,z$ be the fractions carrying neither infection, only $A$, only $B$, and both, with sum one. Let ${\lambda }_A={\beta }_A(x+z)$, ${\lambda }_B={\beta }_B(y+z)$, assume each infection independently recovers at rate $\gamma$, and multiply susceptibility to the second infection by $q>1$. One suitable pair of SIS models isRecovery of one infection from $z$ leads to the other single-infection class, not directly to $s$. The probability simplex is invariant. At the disease-free equilibrium the eigenvalues are ${\beta }_A-\gamma$, ${\beta }_B-\gamma$ and $-2\gamma$. Therefore failure of both isolated and low-density invasion requires ${\mathcal{R}}_A={\beta }_A/\gamma <1$, ${\mathcal{R}}_B={\beta }_B/\gamma <1$. Enhancement of secondary susceptibility does not alter these linear thresholds because coinfection is second order at introduction.

For completeness, a joint positive equilibrium must haveandTo obtain these, divide the equilibrium equations for total $A$ and total $B$ prevalence by their positive prevalences, yielding $s+qy={\mathcal{R}}_A^{-1}$ and $s+qx={\mathcal{R}}_B^{-1}$. Substitution into the $s$ balance gives the quadratic. Its discriminant must be nonnegative and the resulting $x,y,z$ positive. These conditions test equilibrium feasibility for unequal rates; stability must also be checked rather than inferred from the mere presence of a root.

A fully explicit Allee effect results in the symmetric permitted case ${\beta }_A={\beta }_B=\beta$, $\mathcal{R}=\beta /\gamma$. The joint equilibria haveTwo feasible positive equilibria coexist with the stable disease-free state exactly in the strict rangeIndeed at the fold the two roots coalesce, and at $\mathcal{R}=1$ the smaller-prevalence branch reaches the disease-free boundary. For $q\le 2$, roots at $\mathcal{R}<1$ do not give positive singly infected fractions. Strict inequalities exclude these marginal endpoints.

Here is the stability check underlying the cooperative SIS coinfection threshold. On the symmetric subspace put $v=x+z=y+z$ and $c=z$. ThenAt a positive equilibrium $c=q\mathcal{R}{v}^2/(1+q\mathcal{R}v)$. Its two-dimensional Jacobian matrix hasThe higher-prevalence branch has positive determinant and negative trace, while the lower-prevalence branch is a saddle equilibrium. The remaining antisymmetric mode, from $x-y$, has eigenvalue $-q\beta v<0$. Thus the smaller susceptible root $s_{-}$ is a stable joint endemic equilibrium, separated from the stable zero state by a finite seeding threshold. Neither infection can persist alone in this range. Strong cooperation can sustain both infections after a sufficiently large joint introduction even though neither invades from rarity. The numerical condition belongs to the specified secondary-susceptibility model; alternative cooperation mechanisms need their own threshold calculation.

Assume well-mixed populations, exponential prey growth without density regulation, predator mortality without food, and bilinear encounters. The Lotka-Volterra predator-prey model isIts coexistence equilibrium is $X_{\ast }=d/c$, $Y_{\ast }=a/b$. The Jacobian matrix there has eigenvalues $\pm i\sqrt{ad}$, so it is a center equilibrium rather than an attracting equilibrium. More globally,is a first integral: substituting the two equations into $\dot{H}$ gives zero. It has a strict minimum at coexistence, diverges at the boundary and at infinity, and its other positive-quadrant level sets are closed periodic orbits. Small oscillations have frequency $\omega =\sqrt{ad}$. In scaled variables $U=\sqrt{d}(X-X_{\ast })/X_{\ast }$ and $V=\sqrt{a}(Y-Y_{\ast })/Y_{\ast }$, the linearization is $\dot{U}=-\omega V$, $\dot{V}=\omega U$.

Model an annual fixed quota as a pulse removing $h>0$ prey, $X^{+}=X^{-}-h$, $Y^{+}=Y^{-}$, with $h<X^{-}$. Between pulses the original equations apply. In logarithmic coordinates the flow has zero divergence, but the pulse has determinant $X^{-}/(X^{-}-h)>1$. Therefore a periodic annually harvested trajectory has return map determinant greater than one, and cannot have both multipliers inside the unit circle. It is unstable, even when a leading rotation-plus-translation approximation looks neutral. Near a nonresonant small-quota forced orbit this expansion destabilizes the original center equilibrium; resonant pulse timing can add secular effects. Quotas larger than the remaining prey are not biologically realizable and require stopping the model. This is fixed-quota harvesting of Lotka-Volterra populations.

The common continuous approximation to an annual quota gives $\dot{X}=X(a-bY)-h$. Its positive equilibrium is $X_{\ast }=d/c$, $Y_h=(a-h/X_{\ast })/b$, requiring $h<a{X}_{\ast }$. Its Jacobian matrix has trace $h/X_{\ast }>0$ and determinant $bc{X}_{\ast }{Y}_h>0$, so this approximation also destabilizes coexistence. At small $h$ it is an unstable focus. A quota is a larger per-capita mortality when prey are scarce, which is the adverse feedback responsible for the result.

For abundance-triggered pulses, retain the same fixed prey removal and specify the crossing direction. The exact event-triggered harvesting by conserved energy calculation isIf prey abundance triggers the event at $X=X_c$, this increment is the same at every event. A sufficiently high prey threshold with a modest quota gives $\Delta H<0$: the orbit's oscillation energy decreases until it no longer crosses the threshold, after which harvesting stops. If $X_c\le X_{\ast }$, then $\Delta H>0$ and harvesting increases oscillations. At the balanced threshold satisfying $ch=d\log [X_c/(X_c-h)]$, energy is unchanged and the ideal orbit is neutral. Thus the useful rule is harvesting at high prey abundance, not simply using any fixed threshold; the equality case supplies no asymptotic restoration.

A predator threshold $Y=Y_c$ does not hold prey abundance fixed. On a rising-predator crossing, $X>X_{\ast }$, so a sufficiently small harvest initially lowers $H$; on a falling-predator crossing $X<X_{\ast }$, it raises $H$. Nevertheless the rising-predator rule is not a robust stabilizer in this conservative model. Its balanced return orbit has preharvest prey abundanceobtained by equating $cX-d\log X$ before and after the pulse. On the return section the derivative isIts magnitude exceeds one because $e^z-e^{-z}>2z$ for $z>0$. The balanced predator-triggered cycle is consequently unstable. At leading small-oscillation order its multiplier is $-1$, explaining a marginal alternation rather than stable damping. If the trigger is crossed in both directions, or a variable rather than fixed quota is used, a different return rule results and must be analyzed separately.

For conservation, the ideal model shows why fixed removals can be dangerous and why prey-based high-abundance feedback can be safer than a calendar quota or an unspecified predator threshold. These conclusions depend on the chosen pulse rule and on the absence of density dependence, refuges and stochasticity. They do not justify a universal numerical safe quota. A harvest policy needs a stopping threshold and the correct phase-dependent feedback, not merely a small nominal annual catch.