Zooplankton are drifting aquatic consumers, including small animals. In the illustrative food web, the chosen zooplankton feed on algae and serve as prey for small fish; this assumed diet is not a claim about every kind of zooplankton.

Energy flow tracks energy entering an ecosystem, passing through organisms and leaving its boundary. To see its quantitative effect, assume net production at one trophic level is $P_1=1000\mathrm{kJ}{\mathrm{m}}^{-2}\mathrm{da}{\mathrm{y}}^{-1}$ and assume a transfer efficiency $\epsilon =0.10$ at each of two successive steps. Here $P_i$ measures the energy incorporated into new living material per unit area per day, after that level's own cellular respiration; it is not a count of organisms or a standing stock. The joule supplies the energy unit, with $1\mathrm{kJ}=1000\mathrm{J}$.

Only $10/1000=0.01$, or one percent, of the initial production reaches level 3 in this toy food chain. The first step leaves $1000-100=900$ and the second $100-10=90$ outside the next level's new production. These differences can include uneaten material, wastes, and energy released as heat during cellular respiration; their separate amounts have not been assigned. Detritus can still carry energy to decomposers. The 10 percent efficiency is an assumption, not a universal law: transfer efficiencies vary with the organisms, food and conditions. The production ratio $P_{i+1}/P_i$ also differs from the fraction of an individual meal converted into growth. See Clark, Douglas and Choi, Biology 2e, 46.2: production and transfer efficiency.

The shrinking budget explains why supporting another trophic level requires enough production below it. It does not prove an exact maximum length for every food chain, particularly when a food web imports resources or has branching diets. Lindeman's 1942 paper grouped feeding relationships into trophic levels, compared transfers and cellular respiration, and emphasized the ecosystem as a unit connecting living and nonliving components. That viewpoint made different organisms comparable through a common energy budget. See Lindeman, The Trophic-Dynamic Aspect of Ecology (1942).

Material recycling and energy flow describe different bookkeeping. Carbon, nitrogen and phosphorus can be reused through chemical reactions and transfers between organisms and their surroundings. Energy transferred as heat is conserved, but is no longer the same store of chemical energy available to a consumer; photosynthetic primary producers need continuing light input. A decomposer helps recycle material and also releases heat through metabolism. Thus loss from one feeding budget means transfer into other forms or routes, never destruction of energy. See Clark, Douglas and Choi, Biology 2e, 46.3: material cycles. Conservation of energy applies to the whole accounting.

A trophic level describes feeding position relative to primary producers. Assign primary producers level 1; in the illustrated food chain, zooplankton occupy level 2, small fish level 3, and large fish level 4. Following a single feeding path adds one level at each step. The labels make dependence on the initial resource visible, but a branching food web can give an organism several feeding paths. See Clark, Douglas and Choi, Biology 2e, 46.1: feeding positions and branching food webs.

An omnivore can obtain food from more than one trophic level. As an explicitly assumed diet-weighted example, suppose an omnivore gets half its assimilated food energy from level-1 primary producers and half from level-2 consumers. Define its trophic level as one plus the weighted mean level of its food: $1+(1/2)1+(1/2)2=2.5$. This computed position depends on the specified diet and weighting convention; it is not an immutable property of the species. A different life stage or diet can change it.

Detritus is nonliving organic material, including dead remains and organic wastes. It is a resource for decomposers, not itself an organism or a predator. In the illustrative food web, the dashed arrow from algae to detritus denotes a material transfer; the solid arrow from detritus to decomposers denotes resource use.

A consumer obtains organic material by feeding on other organisms or their products. A consumer feeding on primary producers is a primary consumer; one feeding on primary consumers is a secondary consumer. These names describe positions in a food chain, rather than permanent labels for a species.

A primary producer builds organic chemical compounds from inorganic sources. Photosynthetic algae, plants and cyanobacteria use light energy for this role through photosynthesis and carbon fixation. The role supplies new living material to a food web; it does not mean that the organism creates energy.

A hyperbolic equilibrium point has no Jacobian eigenvalue with zero real part. Every linearized mode therefore has an exponentially growing or decaying direction, with no neutral center direction at the linear level. Stable nodes, stable foci and saddles can all be hyperbolic; hyperbolic does not mean attracting.

The absence of neutral eigenvalues makes local classification particularly useful. Small smooth nonlinear terms preserve the qualitative local structure described by the linearization under the hypotheses of the Hartman-Grobman theorem. For example, $\dot{x}=-x$, $\dot{y}=y$ is hyperbolic but unstable because of its growing direction. As a parameter moves an eigenvalue onto the imaginary axis, hyperbolicity fails and nonlinear terms can organize a bifurcation. This identifies where a familiar linear classification needs additional analysis.

A trajectory's limit set consists of its accumulation points along sequences of times tending to positive or negative infinity. The forward-time set is its omega-limit set and the backward-time set its alpha-limit set.

Let $D$ be a simply connected open subset of the plane and let $F$ be a continuously differentiable vector field. If a continuously differentiable multiplier $B$ makes $\nabla \cdot (BF)$ have one sign throughout $D$ and not vanish identically on any nonempty open subset, then $\dot{x}=F(x)$ has no nonconstant periodic orbit contained in $D$.

Indeed, suppose such an orbit bounds a region $R$. Simple connectedness puts its interior in $D$. Along the orbit the vector field is tangent, so $BF$ has zero outward normal flux. Green's theorem would then giveThe sign condition and continuity make the area integral strictly nonzero, a contradiction.

The choice of Dulac function $B$ can expose a sign that the original divergence lacks. Its regularity must hold on the entire simply connected region: a multiplier singular at an enclosed point cannot exclude a cycle enclosing that singularity.

An invariant manifold is a manifold preserved by a dynamical system: a trajectory beginning on it stays on it for as long as that trajectory is defined. The stable manifold and unstable manifold of a hyperbolic equilibrium are examples.

The part of a response that dies away after an initial condition or an input change. In a stable linear ordinary differential equation, the decaying homogeneous solution gives a transient response relative to a chosen particular solution. A homogeneous solution that grows or persists is not a decaying transient.

A differential equation whose evolution rule depends explicitly on time. A time-varying coefficient or forcing term can make a system nonautonomous. Unlike an autonomous differential equation, the state alone need not determine its future without the current time.

An autonomous differential equation has no explicit dependence on time: $\dot{x}=f(x)$. Its current complete state determines the instantaneous rate of change through the same rule at every starting time. A harmonic oscillator written as $\dot{x}=v$, $\dot{v}=-{\omega }_0^{2}x$ is autonomous even though its solution oscillates in time.

With unique solutions, shifting a solution in time gives another solution of the same equation. This allows a phase portrait to be drawn directly in state space, with one vector field assigning a direction at each point. Explicit periodic forcing instead produces a nonautonomous differential equation. One can sometimes recover an autonomous description by adding the drive phase as an extra state variable, but that enlarged state space contains information absent from the original mechanical coordinates alone.

A local transverse section to a flow is mapped back to itself by the next intersection of an orbit. Fixed points represent periodic orbits, and the derivative of this return map determines their transverse stability.

A relative equilibrium is a solution moving only through a continuous group action: its shape is stationary after the symmetry motion is removed. If a one-parameter subgroup is written $g(t)=\exp (t\xi )$, the solution has the form $x(t)=g(t)x_0$. Its image lies on a single group orbit, and quotienting by the symmetry turns it into an ordinary equilibrium point.

For example, the ordinary differential equationis equivariant under $A\mapsto e^{i\alpha }A$. At $\mu >0$, writing $A=r{e}^{i\theta }$ gives $\dot{r}=(\mu -r^2)r$ and $\dot{\theta }=\omega$. Thus $r=\sqrt{\mu }$ is stationary in the quotient, while $A(t)=\sqrt{\mu }{e}^{i(\omega t+{\theta }_0)}$ rotates in the original state space. When $\omega \ne 0$ this is a periodic orbit; when $\omega =0$ each point on the circle is an ordinary equilibrium point.

A relative equilibrium need not be periodic. A steadily translating pattern on an infinite line has a fixed profile in its moving frame but never returns to the same laboratory position. The rotating point-vortex relative equilibrium is the corresponding idea for rigidly rotating vortex configurations.