Population dynamics

A population model has density dependence when its per-capita recruitment, survival or growth depends on population abundance relative to habitat or resource scale. Negative density dependence reduces per-capita growth as crowding increases. In a discrete-generation model, a strong enough decline can make the next total population decrease as the present total increases, producing overcompensation and possible alternating generations. The ecological mechanism and the mathematical stability calculation should be distinguished.

Overcompensation occurs when a larger present population produces a smaller total population in the next generation over part of the relevant range. It is stronger than a decline of per-capita growth alone. For $F(u)=\alpha u/(\beta +u^{\gamma })$ with $\gamma >1$, the large-abundance response $F(u)\sim \alpha u^{1-\gamma }$ decreases with abundance. Stronger feedback can drive a fixed-point multiplier through minus one, but that local threshold alone does not establish chaotic behaviour.

A per-capita mortality rate $a$ means that the death flux from a biological population of size $I$ is $aI$. With constant rate and no other transfers, $\dot{I}=-aI$ gives exponential survival $I(t)=I(0)e^{-at}$. In an epidemic model this can specify mortality from the infective compartment rather than mortality of the entire population.

Constant harvest effort removes individuals at rate proportional to their current population, with fixed coefficient $E$. Its continuous model subtracts $EN$ from the growth rate, while a discrete model may use $N_{t+1}=g(N_t)-E{N}_t$. This differs from constant-quota harvesting, which subtracts a population-independent amount. A discrete rule must remain nonnegative on its physical state range, or require a separately specified truncation.

For $\dot{N}=N[P(N)-E]$ and $P(N)={\prod }_{i=1}^3(a_i-N)$ with $0<a_1<a_2<a_3$, extinction changes stability at $E_T$, while the upper stable population and an unstable separator merge at $E_M$. If $E_T<E_M$, an interval of efforts supports stable extinction and a stable positive population simultaneously; at $E_M$ the positive state collapses. If the thresholds have the opposite order, the high-population pair disappears before the lower stable branch reaches extinction. At a positive equilibrium stability follows from the sign of $N{P}^{\prime }(N)$.

For $g(0)=0$ and strict concavity, per-capita recruitment $g(N)/N$ strictly decreases because $g(N)={\int }_0^N{g}^{\prime }(s)ds>N{g}^{\prime }(N)$. Thus a positive fixed point is unique if it exists. A slope $g^{\prime }(0)>1$ makes extinction unstable but does not guarantee a positive intersection: $g(N)=2N+1-e^{-N}$ stays above the identity. A large-population restriction supplies existence; the local fixed-point stability condition remains $|g^{\prime }(N_{\ast })|<1$.

The rate of addition through birth to a specified biological population. State whether a rate is absolute or per capita and specify its time unit.

A stage-structured population model tracks distinct life stages, such as juveniles and adults, with separate reproduction, survival and transition terms. A matrix update describes the linear case; density-dependent reproduction gives a nonlinear discrete dynamical system.

A carrying capacity is an abundance scale at which a specified environment's net population growth vanishes. In the logistic growth equation, positive unharvested populations approach $K$. It depends on the species, resources and environment rather than being an immutable universal population limit.

The useful distinction is between a growth rate and a sustainable abundance. In a logistic differential equation, $r$ controls how quickly a population responds, while the carrying capacity $K$ sets the positive equilibrium point. Increasing $r$ with $K$ fixed accelerates approach without raising that equilibrium. Increasing $K$ changes the destination. The logistic differential equation diagrams show populations approaching the same carrying capacity from below and above.

In ecological use, the carrying capacity describes what a particular environment can support under specified conditions. Nutrient supply, available space, and accumulated waste can change those conditions; it is not a permanent property of a species alone. A population temporarily above a carrying capacity is possible. In the simplest constant-$K$ logistic growth equation, its net growth becomes negative; real population dynamics can include delays, seasonal variation, and changing resources that the basic mathematical model omits. ^[ref].

The Nicholson-Bailey model describes seasonal host-parasitoid interaction. Hosts surviving random attacks reproduce, while parasitoid offspring are proportional to attacked hosts. For $r>1$, its positive equilibrium is $Y_{\ast }=(\log r)/a$, $X_{\ast }=r{Y}_{\ast }/[c(r-1)]$. Counting attacked rather than surviving hosts makes its predator reproduction law different from the annual pulse-breeding predator-prey model.

At the positive Nicholson-Bailey model equilibrium, $\mathrm{tr}{J}_{\ast }=1+\log r/(r-1)$ lies between one and two, and the displayed determinant exceeds one. These follow from $(r-1)/r<\log r<r-1$ for $r>1$. The discriminant is therefore negative, and the conjugate eigenvalues have moduli greater than one. Nearby annual samples spiral outward rather than approach coexistence.

Two populations grow at the same per-capita rate and each loses members at a rate proportional to their product. On the invariant diagonal $x=y=z$, the logistic differential equation gives $z(t)=\lambda /[\mu +(\lambda -\mu )e^{-\lambda t}]$ for $z(0)=1$ and positive rates. The positive symmetric equilibrium $(\lambda /\mu ,\lambda /\mu )$ is stable along that diagonal but unstable transversely: $(x-y{)}^{\prime }=\lambda (x-y)$, while the Jacobian matrix has eigenvalues $-\lambda ,+\lambda$. Thus an exactly symmetric deterministic solution can relax toward coexistence while a small imbalance grows. A corresponding density-dependent Markov jump process has a controlled fluid limit on each fixed finite interval, rather than uniform control for all time.

The logistic growth equation applies the logistic differential equation to population dynamics. For positive growth rate $r$ and carrying capacity $K$, an initial condition $0<N(0)<K$ gives $N(t)=K/[1+b{e}^{-rt}]$, where $b=K/N(0)-1$. Its per-capita growth falls linearly with abundance, and a trajectory starting below $K/2$ reaches an inflection point when $N=K/2$. The exact solutions of a differential equation and their diagrams are developed under logistic differential equation. A translated and scaled logistic function has the same form. In nonlinear regression for biological size, an analogous saturation curve need not imply that circumference itself obeys a mechanistic population law.

For a fixed catch rate $h$, natural logistic growth can replace the catch at two abundances if $0<h<rK/4$. The upper equilibrium $N_{+}=K[1+\sqrt{1-4h/(rK)}]/2$ attracts, and the lower $N_{-}$ is an unstable survival threshold. They meet in a saddle-node bifurcation at $h=rK/4$, $N=K/2$. Above this catch every positive population declines to zero. Below it a population initially under $N_{-}$ also becomes extinct. A constant catch is stopped at extinction to avoid negative abundances; it is distinct from per-capita harvesting proportional to $N$.