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## Quantitative Biology: populations

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**Quantitative Biology: populations**graham.medley@warwick.ac.uk**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**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**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**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**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**I-D Equilibrium**• When dN/dt = 0 • population rate of change is zero • immigration rate = (population) death rate**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**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**Birth-Death Continuous**• r is the difference between birth and death rates • R = er ; r = ln(R) • If r > 0, exponential growth, if r < 1 exponential decay**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**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)**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**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**Logistic Equation - Empirical**• Empirical observations combined • Fits many, many data**Logistic Equation - Mechanistic**• Linear decrease in per capita birth rate • Linear increase in per capita death rate**Stability of Logistic**• Linear birth and death rates (as functions of N) give a single equilibrium point N = K • Equilibrium is globally, stable**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**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**Multi-population Dynamics**• Two Species • Competition (-/-) • intraspecific • interspecific • Predation (+/-) • patchiness • prey population limitation • multiple equilibria • Multi-species**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**Interspecific Competition**• Competition for shared resource • results in exclusion or coexistence • which depends on degree of overlap for resource and degree of intraspecificcompetition • 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**Interspp Competition Dynamics**• Lotka-Volterra model • Structure • Statics • What are the equilibria • Dynamics • What happens over time • Phase planes • isoclines**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**Analysis**• Equilibrium points are given when the differential equation is zero • A single point (trivial equilibrium) and isocline • The line along which N1 doesn’t change**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…**Dynamics**• Exclusion or co-existence is not dependent on r • but dynamic approach to equilibrium is**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**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)**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 stabilisedynamics**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**Lecture 2: Structuring Populations**• Age • Leslie matrices • Metapopulations • Probability distributions • Metapopulations • Levin’s model**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**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**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**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**Leslie Matrix**• This difference equation can be written in matrix notation**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**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**Spatial Structure**• Many resources are required for life • e.g. plants are thought to have 20-30 resources • light, heat, inorganic molecules (inc. H2O) 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**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**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)**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)