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An annual pulse-breeding predator-prey model combines continuous predation and mortality between short reproduction episodes with an annual difference equation. If surviving predators produce offspring in proportion to prey surviving the year, retained adults give the displayed map. Here $r$ combines prey reproduction and natural survival, $s$ is predator survival, and $\kappa$ integrates predation over declining predator abundance. The positive annual equilibrium is $X_{\ast }=(1-s)/(sb)$, $Y_{\ast }=\log r/\kappa$ for $r>1$ and $0<s<1$.

The goal of ecology is to explain where organisms live, how many there are, and how their interactions with one another and their surroundings change those patterns. Living partners include prey, predators and parasites; nonliving conditions include light, temperature and water. An ecosystem brings these interacting parts into one study system. See Clark, Douglas and Choi, Biology 2e, 44.1: the scope of ecology.

Start with an invented aquatic food web: photosynthetic algae feed zooplankton, which feed small fish, which feed large fish. That food chain has four feeding positions and three feeding arrows. Dead material also reaches decomposers through detritus, so the food web branches rather than forming a single ladder.

The appeal of ecology is that an organism's surroundings include other organisms that change those surroundings. Photosynthesis links light to living material; predation transfers that material between organisms; decomposers return its constituent chemical elements to forms that can be used again. These connections help explain why changing one part of an ecosystem can affect distant parts of its food web, although an arrow alone does not predict the size or even the direction of an indirect change. Population dynamics supplies quantitative tools when the question concerns changing counts.

In 1866, Ernst Haeckel's Generelle Morphologie der Organismen included a section titled Oecologie und Chorologie in volume 2, beginning on page 286. This places the terminology of ecology in an early systematic treatment of organisms and their relationships. See Haeckel, Generelle Morphologie der Organismen, volume 2 (1866), contents and p. 286. A later quantitative landmark is Raymond L. Lindeman's 1942 treatment of ecosystems through feeding relationships and energy flow. See Lindeman, The Trophic-Dynamic Aspect of Ecology, Ecology 23 (1942), 399–417.

A parasitoid develops on or in a host and eventually kills it. Host-parasitoid transmission therefore has different offspring accounting from ordinary predation or a nonlethal parasite-burden model.

Use an annual pulse-breeding predator-prey model. Let $X_n,Y_n$ be the abundances just after reproduction, and let the year have length $T$. Between breeding pulses, assume mass-action predation, natural mortality rates ${\mu }_X,{\mu }_Y>0$, and no reproduction:Solving the predator equation and inserting it into the prey equation gives the pre-breeding abundancesSuppose reproduction multiplies surviving prey by $R$ and adds $q{X}_n^{-}$ offspring per surviving predator. Retaining the surviving adults gives $X_{n+1}=R{X}_n^{-}$, $Y_{n+1}=Y_n^{-}(1+q{X}_n^{-})$. Define $r=R{e}^{-{\mu }_{X}T}$, $s=e^{-{\mu }_{Y}T}$, $\kappa =\beta (1-s)/{\mu }_Y$ and $b=q/R$. The resulting difference equations areThus offspring depend on prey available at the end of the year, not on the number killed during the year. The adult-survival convention is part of the model; if breeding replaces all adults, use $Y_{n+1}=sb{Y}_n{X}_{n+1}$ instead. Both are consistent pulse models under their respective life-history assumptions.

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.

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.

A predator is an organism that consumes prey through predation. A predator need not occupy the highest trophic level of an ecosystem, and the label does not specify its size or guarantee that it has no predators of its own. In the illustrative food chain, the small fish is a predator at level 3 and also prey for the large fish at level 4. Separating these relational labels from species names prevents a misleading division of the world into permanent predator and prey categories.

Prey is an organism consumed by a predator in predation. The name describes the receiving end of that interaction: a small fish can be prey for a large fish while being a predator of zooplankton. In a food web drawn with resource-to-consumer arrows, a feeding arrow goes outward from the prey toward its predator. This convention makes the route of food and energy easy to follow.