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Models of Population Growth • Our models of population growth have assumed that growth rates are constant: – Continuous: r – Discrete: R • Imply I l th thatt populations l ti grow without ith t b bound. d – Unrealistic assumption. – Populations typically limited by available space and resources. Limited population growth • Thomas Malthus (1798): – Populations can increase exponentially. – Food supplies generally increase arithmetically. arithmetically – Population growth will eventually be limited. • Charles Darwin ((1859)) used Malthus’ concept p in formulation of theory of natural selection. – Most organisms produce many more offspring than can survive i tto breed. b d • Estimated that a single pair of elephants would produce nearly 19 million individuals in 750 years. – Competition for limited resources favors individuals better ‘adapted’ to their environment. – Better-adapted individuals produce produce, on average average, more offspring then typical for the population. Limited population growth • Many M ways in i which hi h population l ti growth th might i ht b be lilimited: it d – Shortage of food – Lack of space – Accumulation of toxic substances in the environment • Population increases may increase competition for common resources: – Directly: overt interactions between individuals. – Indirectly: via passive depletion of the environment (interference competition). • Such factors might lead to: – Increased mortality rates – Reduced reproduction rates • Result: population growth rate decreases with increasing population size. Limited population growth • Also Al applies li tto molecular l l and d cellular ll l llevel: l – Chemical reactions: • Increase in concentration of a product may be limited by amount of reactant present. – Nutrient uptake by cell: • May be limited by the concentration in the surrounding medium. • P Processes att allll llevels l llead d tto lilimited it d growth th iin which the growth (or production) rate declines ssteadily ead y with popu population a o ssize e (o (or a amount ou p produced). oduced) • Let’s look at some mathematical models for limited population growth that apply to a wide range of biological processes and mechanisms. Closed-system model for limited growth in continuous time • Basic pattern of limited population growth: – Growth rate declines steadily towards zero zero. – The larger the population, the more slowly it grows. • Monomolecular model: used extensively to describe: – Increase in concentration of a product formed in a 1st-order chemical reaction. – Nutrient uptake by a cell in a constant medium medium. – In both cases, medium is not refreshed. • Consider chemical reaction represented p by y R P. – R = reactant, P = product. – Assumption: concentration of reactant is diminished linearly by the reaction, reaction and is not replaced (closed system) system). – [R] decreases, while [P] increases proportionately. Monomolecular model for limited growth R P. • R Reaction: ti • Assumption: rate of reaction at any instant is proportional to the concentration of reactant remaining at that time. – Let R(t) be concentration of reactant at time t. – Instantaneous rate of decline of R: dR kR dt – As in the model of exponential growth, k is reaction rate constant measured in units of concentration per unit time time. • This is the familiar differential equation for exponential decline,, with solution: R t Ae kt where A (=all) denotes amount of reactant available at time t = 0: R(0). Monomolecular model for limited growth • By definition: R t C t A – R(t) = concentration of reactant at time t. – C(t) = concentration of product at time t. • Then: R t Ae kt A C t Ae kt C t A Ae kt C t A Ae kt C t A 1 e kt • This is the monomolecular growth model. – Growth rate is proportional to resources yet to be achieved. – C(t) is analogous to a population size. Monomolecular model for limited growth • Monomolecular growth model for closed systems: R t C t A dR kt kR k A C R t Ae dt dC C t A 1 e kt kR k A C dt • Extends to cases in which g growth rate declines linearly y with population size. – R(t) is analogous to resources currently available. – C(t) is analogous to a population size. – 2-parameter model: A, k • Also Al kknown as: negative ti exponential ti l growth th model, d l = von Bertalanffy growth model. Monomolecular model for limited growth Conce entration of product [P Populatio on size] • Monomolecular growth curves for various values of the rate constant k: Monomolecular model for limited growth • Ex: Conversion of N-chloroacetanilide into p-chloroacetanilide: – Observed data points and fitted model: Open-system model for limited growth in continuous time • Most basic observed pattern of limited population growth in nature is a sigmoidal (‘S-shaped’) ( S shaped ) curve: Open system: resources replenished, but limited. Logistic model for limited population growth in continuous time • Many different models for sigmoidal growth curves. • Particular case: logistic model: – Widely used in theoretical work. – Empirically found to describe population growth in the field and in laboratory conditions. • To develop: as before, – Set up a differential equation for the instantaneous growth rate. – Solve S l (i (integrate) t t ) the th equation ti to t find fi d the th pattern tt off population size over time. Logistic model for limited population growth • Recall, for exponential growth in continuous time: – Instantaneous growth rate for a population of size N(t) at time t: dN rN (1-parameter model) dt – r is the exponential rate constant, = growth rate. – Assumed to be constant over time time. • Characteristic feature of limited population growth: – Growth rate is not constant over time. – Decreases as the population size increases. – Population size ‘feeds back’ to the growth rate. Logistic model for limited population growth • There are an infinite number of nonlinear models that might be assumed assumed. – Varying numbers of parameters. – More complicated p mechanistic models involve postulates about underlying processes and mechanisms. • Si Simplest l t case: assume that th t the th growth th rate t constant t t (r) declines linearly with increasing population size. – Linearity is a parsimonious assumption: • ‘Assumption of ignorance’, as in linear regression. Logistic model for limited population growth • Rate of change of population size then given by differential equation: dN N N r0 N 1 r0 1 N dt K K – 2-parameter p model: r0, K. – Parameter K is called the carrying capacity of the environment, = equilibrium level. – Parameter r0 called the initial growth rate rate, = intrinsic growth rate, = potential growth rate, = maximum growth rate, when population size is effectively zero. • Usually U ll jjustt called ll d r, nott r0. – When population size is effectively zero, effects of limiting factors are negligible. – As population size N increases, the parenthetical term decreases toward zero. Logistic model for limited population growth • Logistic model: • To solve: Separate variables and integrate: Evaluate integrals: Take exponentials ti l of both sides: Rearrange ea a ge and simplify: dN N rN 1 dt K (instantaneous rate of population change at time t) KdN N K N r dt 1 1 N K N dN r dt log e N log e K N rt c log e N log e K N rt log e N 0 log e K N 0 N 0 e rt N K N K N 0 N t K K rt 1 1 e N 0 ( (also called the Verhulst-Pearl equation, ~1840) Logistic model for limited population growth • Logistic model: dN N rN 1 dt K • Integrated time-dependent form: N t K K rt 1 e 1 N 0 K • Alternate equivalent form: N t r t h 1 e (reparameterized) – Parameter h is the time at which the population reaches half of the carrying capacity: N h 12 K = ‘half-saturation’ constant. – Assumes that initial p population p size N(0) ( ) is less than the carrying capacity K. Logistic model for limited population growth • Properties of the logistic growth curve: (1) The increase of population size with time follows a sigmoidal or S S-shaped shaped curve curve. N=0 and N=K are asymptotes. Logistic model for limited population growth • Properties of the logistic growth curve: (2) The growth rate increases with population size to a peak rate of r K/4 when the population reaches half of carrying capacity. This occurs at a point of inflexion on the logistic curve, where the slope is greatest: Logistic model for limited population growth • Properties of the logistic growth curve: (3) Unlike the case of exponential growth, the logarithm of population size log(N(t)) increases at a decreasing rate: Logistic model for limited population growth • Properties of the logistic growth curve: (4) The log-growth rate decreases linearly with population size size, with slope –rr / K : Logistic model for limited population growth • Properties P ti off the th logistic l i ti growth th curve: (5) Shape of the curve depends on rate constant, r. For practical purposes, the growth curve covers the time span h ± 5/r . h – 5/r h h + 5/r Logistic model for limited population growth • E Ex: G Growth th off a population l ti off yeastt cells ll (Carlson, (C l 1913): – Introduced nutrients and a small number of yeast cells cells. – Number of cells estimated at hourly intervals for 18 hrs. – Logistic g model fitted to data by y nonlinear regression. g Logistic model for limited population growth • Case in which initial population size greater than the carrying capacity: – Logistic g model typically yp y depicts p p population p that increases steadily toward carrying capacity, K. • Never exceeds it. – IIn theory, th could ld ‘‘seed’ d’ an iinitial iti l population l ti size i th thatt exceeds carrying capacity. • Population opu at o ssize e tthen e dec declines es steadily stead y to K. Logistic model has a stable equilibrium at N(t)=K because population l ti returns t th there ffrom any displacement either above or below. A second equilibrium at N(t) = 0 is ‘trivial’, and is unstable.