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Quantitative Biology: populations graham.medley@warwick.ac.uk

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Lecture 1. Basic Concepts & Simplest Models Definitions Basic population dynamics – immigration-death discrete & continuous – birth-death discrete – logistic equation discrete & continuous Multiple species: competition and predation

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Definitions Population – a “closed” group of individuals of same spp. – immigration and emigration rates zero Metapopulation – a collection of populations for which the migration rates between them is defined Community – a closed group of co-existing species

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Fundamental Equation Populations change due to – immigration, emigration additive rates; usually assumed independent of population size – birth, death multiplicative rates; usually dependent on population size

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Immigration-Death (Discrete) Time “jumps” or steps – N is not defined between steps Immigration & death rates constant Death rate is a proportion – the proportion surviving is (1- ) – limits: 0 1

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Immigration-Death (Continuous) Re-expressed in continuous time – N defined for all times Death rate is a per capita rate – the proportion surviving a period of time, T, is exp(- T) – limits: 0

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Immigration-Death Solution

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I-D Equilibrium When dN/dt = 0 – population rate of change is zero – immigration rate = (population) death rate

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Characteristic Timescales Life expectancy, L, determines the timescale over which a population changes (especially recovery from perturbations) L is reciprocal of death rate (in continuous models) In immigration-death model increasing death rate (decreasing life expectancy) speeds progress (decreases time) to equilibrium

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Immigration-death model with different L

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Simplest Discrete Birth-Death Model R is the reproductive rate – the (average) number of offspring left in the next generation by each individual Gives a difference equation – check with fundamental equation Population grows indefinitely if R>1

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Birth-Death Continuous r is the difference between birth and death rates – R = e r ; r = ln(R) If r > 0, exponential growth, if r < 1 exponential decay

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Density Dependence: necessity To survive, in ideal conditions, birth rates must be bigger than death rates – ALL populations grow exponentially in ideal circumstances Not all biological populations are growing exponentially – ALL populations are constrained (birth death) – Density dependence vs. external fluctuations Stable equilibria suggest that density dependence is a fundamental property of populations

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Factors & Processes Density Independence Factors – act on population processes independently of population density Limiting Factors – act to determine population size; maybe density dependent or independent Regulatory Factors – act to bring populations towards an equilibrium. The factor acts on a wide range of starting densities and brings them to a much narrower range of final densities. Density Dependence Factors – act on population processes according to the density of the population – only density dependent factors can be regulatory Factors act through processes to produce effects (eg: drought-starvation- mortality)

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Density Dependent Factors Mechanisms – competition for resources (intra- and interspecific) – predators & parasites (disease) Optimum evolutionary choices for individuals (e.g. group living, territoriality) may regulate population

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Logistic - Observations Populations are roughly constant – K - “carrying capacity” – determined by species / environment combination – density dependent factors Populations grow exponentially when unconstrained – r - intrinsic rate of population change – i.e. before density dependent factors begin to operate r and K are independent

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Logistic Equation - Empirical Empirical observations combined Fits many, many data

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Logistic Equation - Mechanistic Linear decrease in per capita birth rate Linear increase in per capita death rate

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Stability of Logistic Linear birth and death rates (as functions of N) give a single equilibrium point N = K Equilibrium is globally, stable

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Logistic Equation - Dynamics

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Logistic Equation Properties

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Logistic Equation (Discrete) Explicit equilibrium, K Derivation is by considering the relative growth rate from its maximum (1/R) to its minimum (1) The growth rate (R) decreases as population size increases

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Summary Timescales the “system” (population) timescale is determined by the life expectancy of the individuals within the population Density dependence – Birth and death are universal for biological populations – The direct implication is that populations are regulated

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Multi-population Dynamics Two Species – Competition (-/-) intraspecific interspecific – Predation (+/-) patchiness prey population limitation multiple equilibria Multi-species

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Intraspecific Competition Availability of a resource is limited Has a reciprocal effect (i.e. all individuals affected) Reduces recruitment / fitness Consequently produces density dependence Important in generation of skewed distribution of individual quality – Different individuals react differently to competition = creates heterogeneity Inverse dd (co-operation) – Allee Effect

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Interspecific Competition Competition for shared resource – results in exclusion or coexistence which depends on degree of overlap for resource and degree of intraspecific competition Aggregation & spatial effects – disturbance kills better competitor leaving gaps for better colonisers (r- & K- species) – aggregation enhances coexistence “empty” patches allow the worse competitor some space

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Interspp Competition Dynamics Lotka-Volterra model – Structure – Statics What are the equilibria – Dynamics What happens over time – Phase planes isoclines

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Lotka-Volterra Equations Based on logistic equations – One for each species – 21 represents the effect of an individual of species 2 on species 1 – i.e. if 21 = 0.5 then sp. 2 are ½ as competitive, i.e. at individual level interspecific competition is greater than intraspecific competition

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Analysis Equilibrium points are given when the differential equation is zero – A single point (trivial equilibrium) and isocline – The line along which N 1 doesn’t change

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Phase Planes Variables plot against each other Isoclines Direction of change (zero on isocline) – For spp. 1 these are horizontal toward isocline – For spp. 2 these are vertical toward isocline Combine two isoclines and directions on single figure…

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Outcomes K 2 > K 1 / 12 Spp 2 is more competitive at high densities K 2 < K 1 / 12 Spp 2 is less competitive at high densities K 1 > K 2 / 21 Spp 1 is more competitive at high densities Exclusion (initial condition dependent) Spp 1 wins K 1 < K 2 / 21 Spp 1 is less competitive at high densities Spp 2 winsCo-existence

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Dynamics Exclusion or co-existence is not dependent on r – but dynamic approach to equilibrium is

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Predation Consumers – inc. parasites, herbivores, “true predators” predator numbers influenced by prey density which is influenced by predator numbers – circular causality: limit cycles in simple models time delay – in respect of predator population’s ability to grow, r over-compensation – predators effect on prey is drastic

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Predation Dynamics Limit cycles rarely seen heterogeneity in predation – patchiness of prey densities reduced density in prey population – effect ameliorated by reduction in competition (i.e. compensation) increased density in prey population – effect ameliorated by increase in competition (i.e. compensation)

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Refuges Prey aggregated into patches Predators aggregate in prey-dense patches Effect on prey population – prey in less dense patches are most commonly in a partial refuge – they are less likely to be predated Effect is to stabilise dynamics

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Summary Individuals interact with each other – and compete Each individual is affected by the population(s) and each population(s) is affect by the individual – Population dynamics are reciprocal – and reciprocal across level Co-existence is sometimes hard to reproduce in models – How rare is it? Heterogeneity (e.g. patches) tends to enhance co- existence

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Lecture 2: Structuring Populations Age – Leslie matrices Metapopulations – Probability distributions Metapopulations – Levin’s model

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Types of Structuring Individuals in a population are not identical – heterogeneity in different traits trait constant (throughout life) – DNA (with exceptions? e.g. somatic evolution) – gender (with exceptions) trait variable – stage of development, age, infection status, pregnancy, weight, position in dominance hierarchy, etc

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Rate of Change of Structure If trait constant for an individual throughout life, then it varies in the population on time scale of L – e.g. evolutionary time scale; sex ratios If trait variable for an individual, then varies on its own time scale – infection status varies on a time-scale of duration of infectiousness – fat content varies according to energy balance

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Modelling Stages (Discrete) Discrete time model for non-reversible development – at each time step a proportion in each stage die (a proportion s survives) move to next stage (a proportion m) – a number are born, B – complication: s-m

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easiest to chose a time step (which might be e.g. temperature dependent) or stage structure (if not forced by biology) for which all individuals move up

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Leslie Matrix This difference equation can be written in matrix notation

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Properties of Matrix Model No density dependence or limitation – as discrete birth-death process, the population grows or declines exponentially The equivalent value to R is the “dominant eigenvalue” of M – associated “eigenvector” is the stable age distribution If the population grows, there is a stable age distribution – after transients have died away Density dependence can be introduced – but messy

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Leslie Matrix Example This matrix has a dominant eigenvalue of 2 and a stable age structure [ 24 4 1 ] i.e. when the population is at this stable age structure it doubles every time step

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Spatial Structure Many resources are required for life – e.g. plants are thought to have 20-30 resources light, heat, inorganic molecules (inc. H 2 O) etc. Habitats are defined in multi-dimensional space – “niche” is area of suitability in multidimensional space – Areas of differing suitability Disturbance – No habitat will exist forever – Frequency, duration and lethality – Dispersal is a universal phenomena

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Metapopulations A collection of connected single populations – whether a single population with heterogeneous resources or metapopulation depends on dispersal if dispersal is low, then metapopulation degree of genetic mixing human populations from metapopulation to single population? Depends on tempo-spatial habitat distribution & dispersal

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Levins Model Ignore “local” (within patch) dynamics – single populations are either at N=0 or N=K population size – equilibrium points of logistic equation, ignore dynamics between these points (i.e. r)

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Let p be the proportion of patches occupied (i.e. where N=K) – (1-p) is proportion of empty patches – a is rate of extinction (per patch) – m is per patch rate of establishment in empty patch and depends on proportion of patches filled (dispersal)

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Model Results Equilibrium only for m > a – i.e. metapopulation can only exist if local establishment is greater than local extinction Dynamics similar to logistic equation

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Extinction Rates Stochastic probability of extinction – disturbance (not related to population size) – demographic (related to population size - the smaller the population the greater the risk of extinction) Appears to decrease with increasing p Dispersal occurs all the time, tending to increase small populations

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Model with Decreasing Extinction Include dependence of local extinction on total patch occupancy Exponential assumption gives implicit equilibrium result with two possibilities

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Decreasing probability of extinction with increasing proportion of patches occupied is an example of positive density-dependence. The effect here is to create a threshold proportion of patches that need to be occupied to avoid metapopulation extinction. If the metapopulation level effect is due to negative per patch rate of change of patch occupancy at low p, then this is called a metapopulation ‘Allee effect’ (Amarasekare 1998. Allee effects in metapopulation dynamics. American Naturalist. 152). Just as the Levin’s model can be thought of as analogous to the logistic model of population growth rate, if the per capita population growth rate becomes negative at small population size, this creates a threshold population size, below which extinction results – a phenomenon known as the Allee effect (Stephens et al. 1999. What is the Allee effect? Oikos. 87, 185-190).

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Patches as “Networks” Simplest models have all patches equally connected But patches may be connected as networks, for example:

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Networks in Matrix Format This can be written as a matrix of (direct) connections:

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Structure of Connections Multiplying this matrix together once gives the patches connected by two steps etc:

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Summary Heterogeneity between individuals is what biology is about – Extends to heterogeneity between populations Nothing is the same and doesn’t stay the same – We haven’t touched on evolution Dispersal is universal and leads to metapopulations

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Lecture 3. Small Populations Stochastic effects – demographic & environmental – demographic stochasticity in small populations – stochastic modelling e.g. death process; immigration-death process Monte Carlo simulation Probabilities of extinction

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Stochasticity Deterministic models – give expected (average) outcome (in most cases) Demographic – individuals come in single units – e.g. if deterministic model predicts 5.6 individuals, at the limit of accuracy the number can only 5 or 6 Environmental – environments (resource availability) fluctuates “randomly” – chance events (c.f. disturbance)

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Demographic Stochasticity More predominant in small populations – relative error is greater (c.f. plotting on log scale) Gender problems – if a population of 10 individuals produces 15 offspring, there is a 15% chance of 5 or less females

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Pure Death Process Deterministic model – negative exponential decay: N(t)=N(0)exp(- t) Stochastic model – exp(- t) is the probability of an individual surviving to t – if individual survival is independent, then the numbers surviving to t have a binomial distribution with mean N(0)exp(- t)

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Immigration-Death Process Solution is sum of two populations Immigration process is a Poisson process – Poisson distribution for numbers immigrating in a given time Death process – binomial distribution for numbers surviving to given time

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Stochastic Population Processes

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Monte Carlo Simulation More complex models require computer simulation to find solutions Collection of Poisson processes (i.e. assume independence between different processes) Process rates change with time

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Monte Carlo Example Immigration-death process Process rates – immigration: – death: N – total: T = + N Time to next event – from negative exponential distribution with rate parameter T

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Monte Carlo Iteration Calculate time to next event Calculate which event has occurred – /T is probability of immigration – N/T is probability of death Change population Calculate T

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Birth-Death Simulation Deterministic result is the mean of many stochastic results – not true for every model Individual simulations do not “look like” the mean – interpreting data

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Random Walks Alternative view of stochastic models Population size is performing a “random walk” through time Zero is an “adsorbing barrier” (extinction) – Or 1 if dioecious All populations (which have a death process) have a non-zero probability of reaching zero

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Time to Extinction Death Process – mean time to extinction is : Birth-Death Process – for b < d and N 0 = 1 : – depends on the absolute value of b & d (not just difference, r, as deterministic mean) – faster reproducing spp. (big b) have longer times to extinction

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Probability of Extinction Assured in pure death process Birth-Death Process – prob. of extinction by time t : – N 0 lines of descent have to become extinct – prob. of ever extinction (b>d): – ultimate extinction is increasingly unlikely as N 0 increases

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Effect of Increasing Rates Increasing birth rates increases variability Increasing death rates decreases variability In extinction probabilities, increasing variation increases extinction probabilities – Thus, for the same expected growth rate (r and R), increasing birth rates (and increasing death rates) increases chance of eventually reaching absorbing barrier

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Logistic Results Extinction is certain Time to extinction – with N 0 = 1 : – ln(T E ) is a measure of stability of a population

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Summary Dynamics and heterogeneity have a reciprocal effect on each other – Dynamics creates variability – Variability influences dynamics The mean is not always informative – The average human being… Studying variability (and the dynamics of variability) is usually more informative than studying the mean (and the dynamics of the mean)

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Extinction Summary Probability of extinction is less likely – the greater the population size each line of descent has to become extinct – the carrying capacity is large the random walk is further from the absorbing boundary – the greater birth rates are compared to death rates i.e. the larger the value of r – the smaller the variation in population size the smaller the birth rates, but see above Explains why the majority of populations of conservation concern tend to be large mammals in small habitats

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Lecture 4. Modelling

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Why prediction fails Models are necessarily under-specified – They have to be to be useful The correspondence of the causal relationships they embody to actual phenomena is never known to be perfect The observable initial conditions are never perfectly observed There are always unobservable initial conditions. – What if a big asteroid hits? That might be predictable in the sense that the asteroid is already on its collision course with the Earth, but from a practical viewpoint it may be unobservable Parameters are estimates Some processes are chaotic, such that arbitrarily small errors will cumulate to arbitrarily large deviations from prediction

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Principles Exponential growth Density Dependence – positive cooperation; aggregation; Allee effect – negative necessary but not sufficient for stability Circular Causality – pathways created by interaction with environment (inc. other species) – low frequency cycles / oscillations – time delays Limiting Factors – populations exist in complex webs of interaction, but only a few are important at particular times / places

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