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Continue the initial value problem $y^{\prime }=y$, $y(0)=1$, to $t=4$ using the Euler-Heun embedded pair. Choose absolute and relative error tolerances $\mathrm{atol}=10^{-3}$ and $\mathrm{rtol}=0$, so the scalar scale is $s=10^{-3}$ throughout. This intentionally absolute-only test makes the local and accumulated errors directly comparable. Start with $h=0.2$, accept the Heun's method endpoint precisely when $E=|d|/s\le 1$, and after every trial use the capped adaptive step-size controllerFor $E=0$ use the growth factor two. Cap every trial at $4-t_n$; on rejection retain the old time and state. These choices specify an original computation, rather than a claim about the defaults of a particular software solver.

The initial trial has $E=20$ and is rejected. Retrying from $(0,1)$ with $h=0.9(0.2)/\sqrt{20}\approx 0.0402492236$ gives $E=0.81$ and is accepted. The run takes 315 accepted steps and one rejected trial. Increasing exponential growth makes the local error estimator $|d|=|y_n|h^2/2$ grow at a fixed $h$, so subsequent steps shrink roughly like $|y_n{|}^{-1/2}$. The final step is additionally shortened to land at $t=4$.

The right panel plots errors against the accepted-step number, all divided by the same $s$. The actual Heun local defect uses the restarted exact endpoint $y_n{e}^{h_n}$, while actual global error uses $e^{t_{n+1}}$. All accepted $E$ are at most one, and the actual Heun local defects here are even smaller, at most about $0.01143s$. Nevertheless the endpoint is $y_N\approx 54.5833912205$, versus $e^4\approx 54.5981500331$, with absolute error about $0.0147588=14.7588s$. Passing local tests therefore does not certify the original trajectory's global error.

This failure of a global guarantee has an exact explanation for this example. Put $R(h)=1+h+h^2/2$ and $g_n=e^{t_n}-y_n$. ThenEach new local defect is added to the previous error amplified by $e^{h_n}$. Iterating this identity, rather than merely inspecting $E$, accounts for accumulation. The variable-step global ODE error bound describes the corresponding mechanism for more general ordinary differential equations.

Repeating with $\mathrm{atol}=10^{-2}$ gives 98 accepted steps, one rejection and endpoint absolute error about $0.153252$; with $\mathrm{atol}=10^{-4}$ it gives 1001 accepted steps, two rejections and endpoint absolute error about $0.00145642$. Accuracy improves on this smooth problem, but the endpoint error remains about fifteen times the requested local scale. A different initial value problem can amplify or damp defects differently.

For positive recoverable counts $N_0,N_1$ bracketing one growth interval, define its net cell line population doublings byThe formula counts how many factors of two produce the observed net factor. For exactly two divisions per original biological cell, with all daughters recovered, $N_1=2^2{N}_0$, giving $d=2$. Death or loss of biological cells makes this a net estimate of division.^[ref]

Under constant-rate exponential growth, $N(t)=N_0{e}^{rt}$, a growth interval of duration $\Delta t$ givesFor the constructed fourfold increase over $48\mathrm{h}$, $d=2$ and the model doubling time is $24\mathrm{h}$. The proof follows by taking the natural logarithm and solving $e^{r{t}_2}=2$. This is a fitted property of that interval and model: it is not a universal HeLa doubling time, nor the interval between two divisions of each individual biological cell. Across successive growth intervals, add their $d$ values; keep transfer losses separate as in cell culture passage accounting.

Start at one and apply $n$ updates, each multiplying the current value by $1+1/n$. The final value is $(1+1/n{)}^n$: one update gives $2$, two give $2.25$, ten give approximately $2.59374$, and one hundred give approximately $2.70481$. Increasing the number of steps, while reducing each increment proportionally, approaches Euler's number rather than producing unbounded accumulation.

The natural logarithm makes the limit transparent. For $u>0$, its integral definition gives $u/(1+u)\le \ln (1+u)\le u$, by bounding $1/t$ on $[1,1+u]$. With $u=1/n$,The squeeze theorem gives a logarithmic limit of one; continuity of the exponential function gives $(1+1/n{)}^n\to e$. This links a discrete multiplicative rule with continuous exponential growth, without assuming that a finite number of steps equals the limiting value.

The interval required to double a quantity under constant-rate exponential growth. Solving $e^{r{t}_2}=2$ with the natural logarithm gives $t_2=\ln 2/r$. At $r=0.2$ per time unit, it is approximately $3.46574$ time units; every subsequent doubling takes the same interval.

The real exponential function $\exp (x)=e^x$ converts addition into multiplication:Its values are positive, and it defines a bijection from the real numbers to the positive real numbers. The natural logarithm is its inverse function. These properties connect arithmetic with calculus: adding an amount to the input multiplies the output by a fixed factor, and the local rate of change equals the current value. The exponential series gives one construction; an equivalent characterization is the unique solution of a differential equation $y^{\prime }=y$, $y(0)=1$.

For a concrete example, $y(t)=10e^{0.2t}$ starts at $10$ and satisfies $y^{\prime }=0.2y$. The absolute derivative increases with $y$, while the relative rate $y^{\prime }/y$ stays $0.2$. At $t=5$, $y=10e\approx 27.1828$ and $y^{\prime }\approx 5.43656$. The exponential function therefore models accumulation proportional to what is already present, rather than addition of a fixed amount at every step. A positive constant rate gives exponential growth; a negative one gives exponential decay.

Every positive real base $b\ne 1$ fits the same construction through $b^x=e^{x\ln b}$. Thus its derivative is $(\ln b)b^x$: base $e$ is special because $\ln e=1$. The exponential function is also unchanged in form by differentiation and integration, helping explain its repeated appearance in linear ordinary differential equations. Its real definition, inverse relationship, and derivative rules are developed in OpenStax, integrals, exponential functions, and logarithms.

A positive quantity undergoes exponential growth at constant rate $r>0$ when $y^{\prime }=ry$. For $y_0>0$, the solution of a differential equation is $y(t)=y_0{e}^{rt}$. Every interval of duration $\Delta t$ multiplies the value by the same factor $e^{r\Delta t}$, independently of its starting value.

The logistic differential equation is the nonlinear ordinary differential equationAs a mathematical model of population dynamics, it treats abundance $N$ as a continuous nonnegative quantity. The constant $r$ has units of inverse time, while the carrying capacity $K$ has the same units as $N$. The useful idea is to combine growth resembling exponential growth at low abundance with a slowing approach to a finite abundance. The three trajectories below already show an important distinction: starting above the carrying capacity gives a decreasing curve, whereas starting below it gives an increasing one.

For $N>0$, dividing the derivative by abundance gives the instantaneous per-capita rate:It decreases linearly with $N$: it approaches $r$ at low abundance, is $r/2$ at $N=K/2$, vanishes at the carrying capacity, and is negative above it. This per-capita rate differs from the total rate $N^{\prime }$, which also counts how many individuals contribute. It is an instantaneous rate, rather than a fixed percentage increase over a whole day. The linear per-capita assumption and the resulting qualitative picture are developed in MIT 18.03SC's logistic-model notes.

Writing $f(N)=rN(1-N/K)$, the equilibrium points satisfy $f(N)=0$, so the constant solutions of a differential equation are $N(t)=0$ and $N(t)=K$. The phase line points upward for $0<N<K$ and downward for $N>K$. Also $f^{\prime }(0)=r>0$ and $f^{\prime }(K)=-r<0$: the criterion for fixed point stability for an autonomous differential equation makes zero an unstable equilibrium and $K$ an asymptotically stable equilibrium point. Every positive initial condition approaches $K$ as $t\to \infty$; $N_0=0$ stays exactly zero. Instability of zero means that a small positive population grows away from it, not that a zero population spontaneously appears.

For every $N_0>0$, the unique solution of a differential equation for all $t\ge 0$ isThe denominator is positive on this domain, including when $N_0>K$. For $0<N_0<K$, this is a translated and scaled logistic function; its extension to all real times is a sigmoid function. For $N_0>K$ the coefficient of the exponential function is negative, and the future curve instead decreases toward $K$. No nonconstant positive solution of a differential equation crosses $K$ or reaches it at finite time. A short derivation uses the dimensionless variable $z=K/N-1$: the chain rule gives $z^{\prime }=-rz$, so exponential decay yields $z(t)=(K/N_0-1)e^{-rt}$. The substitution divides by $N$, so the zero equilibrium point must be included separately.

The total growth rate is a downward-opening quadratic function of abundance. Completing the square gives its maximum without solving the differential equation:The balance is beautiful: more individuals contribute to growth, but each contributes less, and the strongest total increase occurs halfway to the carrying capacity. This is a maximum over possible abundances. A trajectory starting above $K/2$ never passes through that abundance in its future.

The chain rule connects this maximum to the shape of the time trajectory through its second derivative:For $0<N_0<K/2$, the trajectory initially has convexity and then concavity, with its one future inflection point atFor $N_0=K/2$, this point lies at the initial time, with concavity for $t>0$. For $K/2<N_0<K$, the inflection point belongs to the past extension, so the future trajectory has concavity throughout. Above $K$, the decreasing trajectory has convexity and no inflection point. The two constant solutions of a differential equation have no change of concavity. Thus an entire logistic function has an S shape, but its observed future segment need not show both bends. See also ^[ref].

For the plotted example with $K=1000$, $r=0.4{\mathrm{day}}^{-1}$ and $N_0=100$, time $t$ measured in days givesThe initial total rate is $N^{\prime }(0)=0.4\cdot 100\cdot 0.9=36$ individuals per day, and the initial per-capita rate is $0.36{\mathrm{day}}^{-1}$. The fastest total increase is $rK/4=100$ individuals per day, reached at $t_{1/2}=\ln 9/0.4\approx 5.49306$ days. At that time the per-capita rate has already fallen to $0.2{\mathrm{day}}^{-1}$.

These parameters are illustrative choices, rather than an empirical fit. The equation specifies deterministic evolution under a phenomenological assumption: it summarizes how net growth depends on abundance without separately modelling every process that affects it. The carrying capacity is a parameter for a specified setting, not a universal constant for all populations. In his 1838 paper "Notice sur la loi que la population suit dans son accroissement", Correspondance mathématique et physique 10, pp. 113–121, Pierre-François Verhulst proposed $p^{\prime }=mp-n{p}^2$, equivalent to this equation with $r=m$ and $K=m/n$. He also noted that different assumed growth-retarding terms could describe the observations similarly while giving very different limiting populations. This is an early illustration of why a good-looking curve alone does not establish a unique long-term prediction.^[ref]

The logistic differential equation describes continuous time. The logistic map instead advances in discrete steps and can exhibit a period-doubling cascade and Devaney chaos; the monotone positive trajectories here cannot. Their connection through the forward Euler method is explained under Euler discretization of logistic growth, where a large step can introduce behaviour absent from the original differential equation.

A log-log plot uses logarithmic scales on both axes. For a positive power law $y=C{x}^p$ with $x,C>0$, positive references $x_0,y_0$ giveIts transformed slope is the exponent $p$ when both axes use the same logarithm base and the slope is measured in logarithmic coordinate units. For example, $y=2x^2$ takes the values $2,8,32$ at $x=1,2,4$. Every doubling of $x$ quadruples $y$, giving slope two on a log-log plot.

This differs from a semi-log plot: a power law has constant proportional response to a proportional change of input, whereas exponential growth has constant proportional response to an additive change of input. Both can appear straight with appropriate axes, so one must read the scale labels before interpreting a slope.

The natural logarithm $\ln x$ is the logarithm with base Euler's number, and the inverse function of the real exponential function. A particularly geometric definition isFor $x>1$, it is the area under the hyperbola $y=1/t$ from $1$ to $x$. For $0<x<1$, the reversed integration limits give a negative value. The fundamental theorem of calculus gives $(\ln x{)}^{\prime }=1/x$, so the natural logarithm is increasing, while $(\ln x{)}^{\prime \prime }=-1/x^2$ gives concavity.

The change of variables $t=au$ shows ${\int }_a^{ab}dt/t={\int }_1^{b}du/u=\ln b$ for $a,b>0$. Thus the intervals from $1$ to $2$ and from $2$ to $4$ each contribute area $\ln 2$, explaining $\ln 4=2\ln 2$. Equal ratios give equal increments on a logarithmic scale, even though the ordinary differences grow. This makes the natural logarithm useful for comparing multiplicative changes and solving for time in exponential growth or exponential decay. The integral construction and inverse function properties are discussed in OpenStax, the natural logarithm as an integral.

Because the hazard functions are constant, normalization gives ${\Lambda }_1=2\alpha$ and ${\Lambda }_2=3\alpha$. The population characteristic polynomial factors asThusThese exponential distributions describe ages among currently present biological cells. Their rates differ from the phase-exit rates because exponential growth overrepresents younger entrants. The stable phase ratio is $N_2/N_1=1/2$, so the G1 phase and G2 phase fractions approach $2/3$ and $1/3$.

Increasing $\alpha$ rescales all times by $1/\alpha$: the population undergoes faster exponential growth, its doubling time is $\boxed{(\log 2)/\alpha }$, and the profile means $1/(3\alpha )$ and $1/(4\alpha )$ decrease. The phase fractions do not depend on $\alpha$. The population matrix has exact solutionso population-ratio transients decay relative to the growing term at rate $7\alpha$. This exact population formula does not require initially stationary cell-cycle age densities.

Mid-twentieth-century environmental and population concerns converged around limits to growth, but disagreed over what caused environmental damage and what intervention followed. More people could increase pressure on land, food and water; that did not establish that numbers alone explained depletion or pollution. A strong account follows the transition from resource conservation to chemical contamination and then to explicit global mathematical models, distinguishing observations from conditional projections.

The resource argument was already prominent in 1948. William Vogt's Road to Survival by William Vogt linked human numbers to the productive capacity and degradation of land. Soil loss could reduce future food production, while further population increase enlarged the demands placed on the same resource base. The relevant constraint was not simply the size of the planet: it was the maintenance of the ecological processes on which agricultural production depended. This linked population dynamics with a changing carrying capacity. The argument's policy implications depended on whether one emphasized birth limitation, restoration of land or changes in resource use.^[ref]

The mechanism can be clarified with modern notation. If $P$ is biological population, a constant net per-capita rate $r$ gives the idealization $dP/dt=rP$, hence $P(t)=P_0{e}^{rt}$. A finite food supply cannot support that positive exponential growth indefinitely. But the premise is conditional: observed birth rates, mortality and food production need not remain fixed. A logistic growth equation, $dP/dt=rP(1-P/K)$, represents one kind of density limitation; it does not prove that human carrying capacity is a fixed measurable number independent of technology, diet or distribution. Resource arguments needed empirical work on both population change and the systems supplying people.

The postwar problem was also about the consequences of attempts to sustain human welfare. Increasing agricultural output and suppressing insect-borne disease could reduce mortality or enlarge food supply, while chemical interventions created further ecological effects. Rachel Carson's Silent Spring by Rachel Carson made that feedback concrete. Its Clear Lake example followed DDD through plankton, fish and fish-eating birds: small environmental concentrations could become large tissue residues through exposure and feeding relationships. The account connected intended pest control with unintended damage to other biological populations. It did not depend on proving that the number of human inhabitants alone had caused the damage.^[ref]

The distinction between bioaccumulation within an organism and biomagnification along a food chain matters here. Neither follows simply from dividing a pesticide dose by the number of people. Persistence, uptake, elimination and feeding relations determine exposure. Rachel Carson's 1963 response to critics listed continuing wildlife losses under officially organized spraying and protests by farmers themselves. The historical conflict was therefore over the evidential adequacy and distribution of an intervention's costs, not merely a choice between caring about people and caring about nature.^[ref]

Paul R. Ehrlich and John P. Holdren's 1971 paper explicitly defended the importance of population growth while addressing consumption and technology. Its decomposition of total impact into numbers of people and impact per person also made clear why density alone is inadequate. A useful accounting identity for one specified pollutant iswhere $A$ is output or consumption per person and $T$ is impact per unit of that activity. For a waste stream measured in mass per year, the units multiply consistently to mass per year. This three-factor notation is an analytical extension of the debate, not a quotation of every equation in the 1971 article. It identifies separate questions about abundance, consumption and production techniques; it is not by itself an independently tested causal law.^[ref]

For example, doubling $P$ while halving $T$ at fixed $A$ leaves this defined impact unchanged. Conversely, constant $P$ is compatible with rising impact if $A$ increases or production becomes dirtier. These counterexamples do not show that population is irrelevant. They show why an explanation must state what else is changing. They also explain a political disagreement within environmentalism: concentrating on fertility can obscure high consumption and harmful production, while concentrating exclusively on cleaner technology can overlook the scale of demand.

The Limits to Growth by Donella H. Meadows and others in 1972 joined population, industrial production, food, resources and pollution in a computer computer simulation. Its importance lay in their interaction and in delays: a system could overshoot a sustainable level before feedback sufficiently changed investment or demographic behavior. The resulting curves were conditional scenarios generated by assumptions about those relations, not direct observations of the future. The authors examined changed assumptions and possible stabilization; the report should not be summarized as one unconditional prediction of a uniquely dated collapse.^[ref]

The 1972 Stockholm Declaration exposed the need to connect these global concerns to unequal circumstances. Principle 8 treated economic and social development as necessary to environmental improvement, while principle 16 discussed demographic policies subject to human rights and governmental judgment. Their coexistence matters: a global population total did not erase differences in poverty, consumption or responsibility.^[ref]

Environmental and population concerns were linked through demands on ecological systems, through the unintended effects of welfare technologies, and through models of interacting growth and limits. They never established one agreed diagnosis. The enduring historical issue was which variables, mechanisms and distributions of responsibility an apparently simple claim about too many people concealed.

A material interface obeys $D(y-\eta (x,t))/Dt=0$. In each fluid this gives ${\varphi }_{jy}={\eta }_t+(U_j+{\varphi }_{jx}){\eta }_x$ at $y=\eta$, where $U_1=U$ and $U_2=V$. Discarding products of perturbations and evaluating at the undisturbed interface givesFor an inviscid irrotational fluid, unsteady Bernoulli is ${\Phi }_t+\frac{1}{2}|\nabla \Phi {|}^2+p/\rho +gy=C(t)$, with total potential $\Phi =U_{j}x+{\varphi }_j$. Subtract the undisturbed relation and linearize at the moving interface. The pressure perturbation there is $-{\rho }_j({\varphi }_{jt}+U_j{\varphi }_{jx}+g\eta )$. With no surface tension the two pressures agree, givingUse complex amplitudes and take real parts at the end. For $k>0$, solutions of Laplace's equation decaying away from the interface have the formPut ${\Omega }_1=\omega -Uk$ and ${\Omega }_2=\omega -Vk$. The kinematic conditions imply $B_1=-i{\Omega }_{1}A/k$ and $B_2=i{\Omega }_{2}A/k$. Substituting into the pressure condition givesA nonzero amplitude therefore requiresTo determine stability, let $R={\rho }_1+{\rho }_2$ and $\overline{U}=({\rho }_{1}U+{\rho }_{2}V)/R$. Completing the square yieldsFor $U\ne V$, the right-hand side is negative exactly whenThen the two frequencies have opposite nonzero imaginary parts. The root with positive imaginary part gives exponential growth through $e^{-i\omega t}$. In terms of wavelength $\lambda =2\pi /k$, the unstable range isThis is the short-wavelength Kelvin-Helmholtz instability for a density-stable interface without surface tension. Equality is the neutral threshold; if $U=V$, the gravitational term is positive and this shear instability is absent.

Let $\Sigma ={\rho }_1+{\rho }_2$, $\Delta ={\rho }_2-{\rho }_1>0$. Completing the square in the dispersion relation givesThe coefficients are real. There is an exponentially growing temporal mode exactly when the right side is negative, because one root then has positive imaginary part in $e^{-i\omega t}$. Rearrangement givesEquality is the marginal threshold rather than exponential growth.

Here $a,b\ge 2$ are fixed integers. The term $a{b}^n$ has exponential growth, whereas $a{n}^b$ has fixed-degree polynomial growth and $n{b}^a$ is linear. Since a fixed-base exponential eventually dominates every fixed-degree polynomial,Hence the requested Big O notation bound, with its matching lower order, is $\boxed{\Theta (b^n),\text{in particular }O(b^n)}$.

False. Put $r_n=3^n/n^3$ for $n\ge 1$. For $n\ge 4$,Thus $r_n\ge r_4(192/125{)}^{n-4}\to \infty$. No fixed constant can bound $3^n/n^3$ eventually, so $3^n\ne O(n^3)$. This is an explicit comparison of exponential growth with polynomial growth.

For a stipulated population of amplifiable copies, let $N_n$ be the amount after cycle $n$ and define PCR efficiency by the fractional new yieldHere $E_n=1$ means one additional copy per existing copy, hence ideal doubling. If every cycle has the same PCR efficiency $E$, the exponential growth model gives $N_n=N_0(1+E{)}^n$. Starting with $N_0=100$ copies, twenty ideal cycles give $104857600$ copies; the stipulated value $E=0.8$ gives about $12748236$. An efficiency of 80 percent therefore means multiplication by 1.8 each cycle, not retention of 80 percent of the final ideal yield.

In the ideal strand-counting example, all duplex-equivalent material doubles, but exact-length amplicons lag initially. Starting from one longer duplex and retaining its two original strands, cycle $n\ge 1$ contains $2(2^n-n-1)$ exact-length strands out of $2^{n+1}$ total strands. Thus their fraction is $1-(n+1)/2^n$: zero after cycle 1, one quarter after cycle 2 and one half after cycle 3. This counts strands rather than assuming every strand immediately finds an exact-length partner. The generic yield model above ignores this early distinction.

A semi-log plot uses a logarithmic scale on one axis and a linear scale on the other. Which axis is logarithmic matters. Plotting $x$ against $\log y$ makes positive exponential functions appear as straight lines; plotting $\log x$ against $y$ instead makes an affine function of a logarithm appear straight. The plotting operations change positions while retaining the original tick values.^[ref]

Take the original dimensionless mathematical model $y(t)=3e^{0.4t}$ for $0\le t\le 8$. It has exponential growth, with $y(0)=3$, $y(4)\approx 14.8591$, and $y(8)\approx 73.5976$. Taking a base-ten logarithm givesThe slope in these transformed positions is $0.4/\ln 10\approx 0.173718$, although the original relative growth rate $y^{\prime }/y$ is $0.4$. Equal steps of two in $t$ multiply $y$ by $e^{0.8}\approx 2.22554$. The corresponding ordinary gains increase, so the left plot bends while the right plot is straight.

One practical use is checking a proposed exponential distribution for failure times. For rate $\lambda >0$ and $t\ge 0$, its cumulative distribution function is $F(t)=1-e^{-\lambda t}$, so the reciprocal survival function $y=1/[1-F(t)]$ satisfies ${\log }_{10}y=\lambda t/\ln 10$. Estimated probabilities can therefore be placed on a logarithmic vertical axis and compared with a straight line. Agreement supports compatibility with the chosen statistical model; a roughly straight finite sample does not establish a unique physical mechanism.^[ref]

For frequency-dependent transmission, the closed SIR model is a system of ordinary differential equations for the fractions $s,i,r$:Here $\beta >0$ is the coefficient of frequency-dependent transmission, and $\gamma >0$ is the recovery rate, interpreted more generally as the removal rate. Both have units of inverse time. With a constant recovery rate, the associated individual time in the infectious compartment has an exponential distribution with mean $1/\gamma$; the SIR model does not require every person to spend exactly that long there. The fractions themselves have no units.^[ref]

For counts $S=Ns$, $I=Ni$, $R=Nr$, the same SIR model equations readThe factor $1/N$ keeps the same $\beta$ in both descriptions. Dropping it while keeping these units would fail dimensional analysis. Some existing SIR model variants use the unnormalized mass-action infection convention instead, including the SIR susceptible-recovered identity; their coefficient is $b=\beta /N$ in the notation used here.

Adding the SIR model equations gives $d(s+i+r)/dt=0$, a short proof of conservation of population. Nonnegative starting fractions stay nonnegative: the outflow from either the susceptible compartment or the infectious compartment vanishes when that group is empty, while $dr/dt\ge 0$. Thus the susceptible compartment can only shrink and the removed compartment can only grow.

For a concrete initial value problem, stipulateAt the start, the flow into the infectious compartment is $0.3\times 0.99\times 0.01=0.00297$ of the total per day, and the flow into the removed compartment is $0.1\times 0.01=0.00100$ of the total per day. Hence $di/dt=0.00197$ per day. If $N=1000$, these correspond to model rates of $2.97$ and $1.00$ people per day, and net growth $1.97$ people per day. A rate need not be an integer count of events on any particular day.

For this SIR model, the basic reproduction number isIt describes one introduced member of the infectious compartment in an otherwise wholly susceptible compartment. The instantaneous reproduction number at the actual starting state is $R_0{s}_0=2.97$, obtained by freezing the current conditions and the current susceptible compartment's fraction. More generally it is $R_{0}s(t)$ in this SIR model. The basic reproduction number is a ratio of two rates, so it has no units; $r_0$ is a different quantity, the initial removed compartment's fraction.

While $i>0$, its instantaneous relative growth isInitially this equals $0.3\times 0.99-0.1=0.197{\mathrm{day}}^{-1}$. Freezing $s\approx s_0$ gives the early exponential growth approximation $i(t)\approx 0.01e^{0.197t}$ with $t$ measured in days. It must fail as the susceptible compartment shrinks. An epidemic invasion threshold occurs at $R_0{s}_0=1$; if $i_0=0$, the deterministic SIR model stays at $i=0$ even above that threshold.

The specific growth rate is biomass production per unit existing biomass per unit time. For biomass concentration $x>0$ with growth alone, $\dot{x}=\mu x$ and $\mu =\mathrm{d}(\ln x)/\mathrm{d}t$. A constant positive specific growth rate produces exponential growth, $x(t)=x(0)e^{\mu t}$, with doubling time $\ln 2/\mu$. In a chemostat, removal makes the net rate $\dot{x}/x=\mu (s)-D$, rather than $\mu (s)$ itself.