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For $A,B,\rho >0$, these Lotka-Volterra equations describe two populations limited by their own density and depressed by the other population. The nonnegative quadrant is invariant. Positive equilibria are intersections of the non-axis nullclines $u+Av=a$ and $Bu+v=b$; their existence and stability distinguish stable coexistence, competitive exclusion, and bistability. Positive solutions have no periodic orbits: multiplying the vector field by the Dulac function $1/(uv)$ makes its divergence $-1/v-\rho /u<0$, so the Bendixson-Dulac theorem applies. Boundedness, boundary equilibria, and their stable manifolds must still be checked before concluding convergence to one particular equilibrium.

For an unregulated exponential basal resource and nearest-neighbor Lotka-Volterra equations, the weighted interaction matrix is skew-symmetric matrix. An even chain has an invertible tridiagonal interaction matrix; its positive equilibrium, when feasible, fixes all long-term means. An odd chain has a null direction and requires a special balance between resource growth and mortality to permit bounded full coexistence. For a three-level chain, $X_1^{b_2}{X}_3^{a_1}=C\exp [(b_{2}r-a_1{d}_3)t]$. A positive even-chain equilibrium gives a conserved coercive relative-entropy-type function, proving every species remains bounded above and away from zero.

Let $X,Y$ be prey and predator abundances. With constant positive demographic and predation coefficients, the Lotka-Volterra equations areTheir positive equilibrium is $X_{\ast }=\gamma /\delta$, $Y_{\ast }=\alpha /\beta$. Its Jacobian matrix and eigenvalues areThe linearized populations oscillate with period $2\pi /\sqrt{\alpha \gamma }$, with the predator peak following the prey peak. This is a center equilibrium, not an attracting equilibrium. The conclusion is supported beyond linearization by the Logarithmic first integral of the Lotka-Volterra equations: in these units one may useDirect differentiation gives $\dot{V}=0$. Its Hessian matrix is a positive-definite matrix in the positive quadrant, and its minimum is at $(X_{\ast },Y_{\ast })$. Nearby level curves are closed periodic orbits, so there is no damping toward a preferred oscillation amplitude.

Not in general. A food web with Lotka-Volterra equations has per-capita balances ${\dot{X}}_i/X_i=r_i+{\sum }_j{A}_{ij}{X}_j$. If all species remain bounded above and away from zero, averaging requires $r+A\overline{X}=0$ with positive means. Many webs have no feasible positive solution. Even a feasible solution does not ensure stability or persistence when the weighted skew-symmetric matrix structure and its coercive first integral are absent.

A simple resource shared by two consumers already illustrates competitive exclusion principle. Suppose their per-capita equations are $\dot{Y}/Y=cX-d$ and $\dot{Z}/Z=eX-f$. ThenIf $d/c\ne f/e$, the ratio has exponential drift, incompatible with bounded persistence of both consumers. The lower break-even-resource consumer has the advantage; equal break-even requirements are a special neutral balance, not generic coexistence. More complicated feeding links can provide additional niches, but positive coefficients and a larger web do not by themselves guarantee that every species survives. One must test feasibility, invasion into the relevant boundary states and stability, with explicit assumptions about resource renewal and self-limitation.

Use a basal resource $X_1$, herbivore $X_2$ and predator $X_3$, with positive encounter, conversion and mortality parameters:Only the resource has intrinsic positive growth; the consumers die without food. There is no resource carrying capacity, immigration or feeding on nonadjacent levels. These assumptions define the simplest food chain with Lotka-Volterra equations.

A positive equilibrium would require both $X_2=r/a_1$ and $X_2=d_3/b_2$. Hence three-level coexistence is possible only on the special balance $\kappa =b_{2}r/a_1-d_3=0$. The exact identityfollows by combining $b_{2}d\log X_1/dt$ and $a_{1}d\log X_3/dt$. It gives a useful classification rather than incorrectly postulating a generic coexistence fixed point.

If $\kappa <0$, the top predator cannot invade the underlying resource-herbivore cycle: its average per-capita growth there is $\kappa$. This extinction statement can be strengthened. The boundary equilibrium has $X_{1\ast }=d_2/b_1$, $X_{2\ast }=r/a_1$, $X_{3\ast }=0$. With weights $w_1=1$, $w_2=a_1/b_1$, $w_3=a_1{a}_2/(b_1{b}_2)$, defineDirect cancellation of paired trophic terms gives $\dot{V}=w_3\kappa X_3<0$. Its level sets bound the first two species above and away from zero and bound $X_3$. Integration and uniform continuity imply $X_3\to 0$; the limiting dynamics are the usual resource-herbivore closed orbits, not generally an attracting equilibrium.

If $\kappa =0$, there is a line of positive equilibria: $X_2=r/a_1$, $b_1{X}_1-a_2{X}_3=d_2$. Choosing any positive point on that line gives a conserved coercive version of $V$ with all three relative-entropy terms. Together with the product integral, it confines generic trajectories to closed curves, with neutral oscillations around the equilibrium line. All three can persist, but only under this finely balanced parameter condition.

If $\kappa >0$, the product identity grows without bound, so at least one of $X_1,X_3$ must become unbounded in the unregulated model. There is no bounded persistent three-species attractor. Finite-time blow-up is excluded by a weighted total-abundance bound whose encounter terms cancel and whose derivative is at most a constant times that abundance. The unbounded long-term behavior signals the missing biological resource regulation. The simplest odd chain does not generically permit bounded full coexistence; adding resource self-limitation changes this conclusion.

Predation is the consumption of prey by a predator, affecting the demographic rates of both populations.

In the usual animal-feeding sense, a predator kills and eats its prey. This distinguishes the interaction from feeding on already-dead material and from the prolonged exploitation of a living host in parasitism. In the illustrative food web, a small fish eating zooplankton is a predator; a large fish eating that small fish makes the same small fish its prey. These roles belong to a particular interaction. See Clark, Douglas and Choi, Biology 2e, 45.6: predation and community interactions.

For a toy count, suppose 50 small fish begin a day and 8 are eaten by large fish. With births, other deaths and movement explicitly omitted, 42 small fish remain, and the fraction removed by predation is $8/50=0.16$. This measures removal of prey, not production of new predators. Food intake can support survival and reproduction, but eight meals do not imply eight new predators; energy flow and life-history details intervene. Lotka-Volterra equations provide one idealized account of such population dynamics, rather than a guarantee that every interacting pair must cycle.