2 Lecture 1. Basic Concepts & Simplest Models DefinitionsBasic population dynamicsimmigration-deathdiscrete & continuousbirth-deathdiscretelogistic equationMultiple species: competition and predation
3 Definitions Population Metapopulation Community a “closed” group of individuals of same spp.immigration and emigration rates zeroMetapopulationa collection of populations for which the migration rates between them is definedCommunitya closed group of co-existing species
4 Fundamental Equation Populations change due to immigration, emigration additive rates; usually assumed independent of population sizebirth, deathmultiplicative rates; usually dependent on population size
5 Immigration-Death (Discrete) Time “jumps” or stepsN is not defined between stepsImmigration & death rates constantDeath rate is a proportionthe proportion surviving is (1- )limits: 0 1Compare this equation with the fundamental equation. Note that the rates are not given the same notation, because it is the concept, not the symbol its given, which is important.
6 Immigration-Death (Continuous) Re-expressed in continuous timeN defined for all timesDeath rate is a per capita ratethe proportion surviving a period of time, T, is exp(-T)limits: 0dN/dt is the rate of change of the population (numbers per unit time), and is called a differential. Consequently, this is a differential equation. It can be zero (which represents an equilibrium or unchanging population), positive (which implies an increasing population) or negative (a decreasing population). It is useful to think of dN/dt as a direction (up or down) as the population moves through time - the larger the size of dN/dt, the larger the slope.
7 Immigration-Death Solution The immigration-death process in continuous time has the solution given below. It depends on the initial conditions, N(0) - which you need if you have a set of directions (the differential equation). Note that the solution is the sum of two populations: the original (initial) population and the new immigrants into the population. This equation is solved in the spreadsheet “Immigration Death”, and you should be able to see it by clicking on the title (when in slide show view).
8 I-D Equilibrium When dN/dt = 0 population rate of change is zero immigration rate = (population) death rateThere is only one equilibrium, when the two rates are equal.
9 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 equilibriumFor example, removing all 19 year old people from a population now, would have an effect on the human population for about 50 years (see later in course), whereas removing all 3 week old mice would not be observable in 2 years time.
11 Simplest Discrete Birth-Death Model R is the reproductive ratethe (average) number of offspring left in the next generation by each individualGives a difference equationcheck with fundamental equationPopulation grows indefinitely if R>1The population 1) stays constant if R=1; 2) grows geometrically or exponentially (i.e. the rate of increases as the population increases) if R>1; 3) decreases exponentially (i.e. the rate of decreases slows as the population decreases) if R<1. This is clearly not realistic as most populations have some equilibrium.
12 Birth-Death Continuous r is the difference between birth and death ratesR = er ; r = ln(R)If r > 0, exponential growth, if r < 1 exponential decay
13 Density Dependence: necessity To survive, in ideal conditions, birth rates must be bigger than death rates ALL populations grow exponentially in ideal circumstancesNot all biological populations are growing exponentially ALL populations are constrained (birth death)Density dependence vs. external fluctuationsStable equilibria suggest that density dependence is a fundamental property of populations
14 Factors & Processes Density Independence Factors Limiting Factors act on population processes independently of population densityLimiting Factorsact to determine population size; maybe density dependent or independentRegulatory Factorsact 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 Factorsact on population processes according to the density of the populationonly density dependent factors can be regulatoryFactors act through processes to produce effects (eg: drought-starvation-mortality)
15 Density Dependent Factors Mechanismscompetition for resources (intra- and interspecific)predators & parasites (disease)Optimum evolutionary choices for individuals (e.g. group living, territoriality) may regulate population
16 Logistic - Observations Populations are roughly constantK - “carrying capacity”determined by species / environment combinationdensity dependent factorsPopulations grow exponentially when unconstrainedr - intrinsic rate of population changei.e. before density dependent factors begin to operater and K are independent
17 Logistic Equation - Empirical Empirical observations combinedFits many, many data
18 Logistic Equation - Mechanistic Linear decrease in per capita birth rateLinear increase in per capita death rate
19 Stability of LogisticLinear birth and death rates (as functions of N) give a single equilibrium pointN = KEquilibrium is globally, stable
22 Logistic Equation (Discrete) Explicit equilibrium, KDerivation is by considering the relative growth rate from its maximum (1/R) to its minimum (1)The growth rate (R) decreases as population size increasesThe derivation of this equation comes from BMT page 53, and is done by knowing two end points on a straight line relationship between the population size and the rate of change between time steps.
23 Because birth and death dominate population processes (by definition), and are multiplicative (i.e. populations tend to double or halve rather than change by a fixed number) the changes in population are best plotted using logarithms.The upper graph shows the discrete logistic curve (as lower graph), and the discrete birth-death model (with R>1) for comparison. Note that the geometric increase appears as a straight line on a logarithmic scale.
24 Summary Timescales Density dependence the “system” (population) timescale is determined by the life expectancy of the individuals within the populationDensity dependenceBirth and death are universal for biological populationsThe direct implication is that populations are regulated
25 Multi-population Dynamics Two SpeciesCompetition (-/-)intraspecificinterspecificPredation (+/-)patchinessprey population limitationmultiple equilibriaMulti-species
26 Intraspecific Competition Availability of a resource is limitedHas a reciprocal effect (i.e. all individuals affected)Reduces recruitment / fitnessConsequently produces density dependenceImportant in generation of skewed distribution of individual qualityDifferent individuals react differently to competition = creates heterogeneityInverse dd (co-operation)Allee Effect
27 Interspecific Competition Competition for shared resourceresults in exclusion or coexistencewhich depends on degree of overlap for resource and degree of intraspecific competitionAggregation & spatial effectsdisturbancekills better competitor leaving gaps for better colonisers (r- & K- species)aggregation enhances coexistence“empty” patches allow the worse competitor some space
28 Interspp Competition Dynamics Lotka-Volterra modelStructureStaticsWhat are the equilibriaDynamicsWhat happens over timePhase planesisoclines
29 Lotka-Volterra Equations Based on logistic equationsOne for each species21 represents the effect of an individual of species 2 on species 1i.e. if 21 = 0.5 then sp. 2 are ½ as competitive, i.e. at individual level interspecific competition is greater than intraspecific competition
30 AnalysisEquilibrium points are given when the differential equation is zeroA single point (trivial equilibrium) and isoclineThe line along which N1 doesn’t change
31 Phase Planes Variables plot against each other Isoclines Direction of change (zero on isocline)For spp. 1 these are horizontal toward isoclineFor spp. 2 these are vertical toward isoclineCombine two isoclines and directions on single figure…
32 (initial condition dependent) OutcomesK2 > K1/12Spp 2 is more competitive at high densitiesK2 < K1/12Spp 2 is less competitive at high densitiesK1 > K2/21Spp 1 is more competitive at high densitiesExclusion(initial condition dependent)Spp 1 winsK1 < K2/21Spp 1 is less competitive at high densitiesSpp 2 winsCo-existence
33 Dynamics Exclusion or co-existence is not dependent on r but dynamic approach to equilibrium is
35 PredationConsumersinc. parasites, herbivores, “true predators”predator numbers influenced by prey density which is influenced by predator numberscircular causality: limit cycles in simple modelstime delayin respect of predator population’s ability to grow, rover-compensationpredators effect on prey is drastic
36 Predation Dynamics Limit cycles rarely seen heterogeneity in predation patchiness of prey densitiesreduced density in prey populationeffect ameliorated by reduction in competition (i.e. compensation)increased density in prey populationeffect ameliorated by increase in competition (i.e. compensation)
37 Refuges Prey aggregated into patches Predators aggregate in prey-dense patchesEffect on prey populationprey in less dense patches are most commonly in a partial refugethey are less likely to be predatedEffect is to stabilise dynamics
38 Summary Individuals interact with each other and competeEach individual is affected by the population(s) and each population(s) is affect by the individualPopulation dynamics are reciprocaland reciprocal across levelCo-existence is sometimes hard to reproduce in modelsHow rare is it?Heterogeneity (e.g. patches) tends to enhance co-existence
39 Lecture 2: Structuring Populations AgeLeslie matricesMetapopulationsProbability distributionsLevin’s model
40 Types of Structuring Individuals in a population are not identical heterogeneity in different traitstrait constant (throughout life)DNA (with exceptions? e.g. somatic evolution)gender (with exceptions)trait variablestage of development, age, infection status, pregnancy, weight, position in dominance hierarchy, etc
41 Rate of Change of Structure If trait constant for an individual throughout life, then it varies in the population on time scale of Le.g. evolutionary time scale; sex ratiosIf trait variable for an individual, then varies on its own time scaleinfection status varies on a time-scale of duration of infectiousnessfat content varies according to energy balance
42 Modelling Stages (Discrete) Discrete time model for non-reversible developmentat each time step a proportion in each stagedie (a proportion s survives)move to next stage (a proportion m)a number are born, Bcomplication: s-m
43 easiest to chose a time step (which might be e. g 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
44 Leslie MatrixThis difference equation can be written in matrix notation
45 Properties of Matrix Model No density dependence or limitationas discrete birth-death process, the population grows or declines exponentiallyThe equivalent value to R is the “dominant eigenvalue” of Massociated “eigenvector” is the stable age distributionIf the population grows, there is a stable age distributionafter transients have died awayDensity dependence can be introducedbut messy
46 Leslie Matrix ExampleThis matrix has a dominant eigenvalue of 2 and a stable age structure [ ]i.e. when the population is at this stable age structure it doubles every time step
47 Spatial Structure Many resources are required for life e.g. plants are thought to have resourceslight, heat, inorganic molecules (inc. H2O) etc.Habitats are defined in multi-dimensional space“niche” is area of suitability in multidimensional spaceAreas of differing suitabilityDisturbanceNo habitat will exist foreverFrequency, duration and lethalityDispersal is a universal phenomena
48 Metapopulations A collection of connected single populations whether a single population with heterogeneous resources or metapopulation depends on dispersalif dispersal is low, then metapopulationdegree of genetic mixinghuman populations from metapopulation to single population?Depends on tempo-spatial habitat distribution & dispersal
49 Levins Model Ignore “local” (within patch) dynamics single populations are either at N=0 or N=K population sizeequilibrium points of logistic equation, ignore dynamics between these points (i.e. r)
50 Let p be the proportion of patches occupied (i.e. where N=K) (1-p) is proportion of empty patchesa is rate of extinction (per patch)m is per patch rate of establishment in empty patch and depends on proportion of patches filled (dispersal)
51 Model Results Equilibrium only for m > a i.e. metapopulation can only exist if local establishment is greater than local extinctionDynamics similar to logistic equation
53 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 pDispersal occurs all the time, tending to increase small populations
54 Model with Decreasing Extinction Include dependence of local extinction on total patch occupancyExponential assumption gives implicit equilibrium result with two possibilities
57 Decreasing probability of extinction with increasing proportion of patches occupied is an example of positive density-dependence. The effect here isto create a threshold proportion of patches that need to be occupied toavoid metapopulation extinction.If the metapopulation level effect is due to negative per patch rate ofchange of patch occupancy at low p, then this is called a metapopulation‘Allee effect’ (Amarasekare Allee effects in metapopulation dynamics.American Naturalist. 152). Just as the Levin’s model can be thought of asanalogous to the logistic model of population growth rate, if the per capitapopulation growth rate becomes negative at small population size, thiscreates a threshold population size, below which extinction results– a phenomenon known as the Allee effect (Stephens et al What isthe Allee effect? Oikos. 87, ).
58 Patches as “Networks”Simplest models have all patches equally connectedBut patches may be connected as networks, for example:
59 Networks in Matrix Format This can be written as a matrix of (direct) connections:
60 Structure of Connections Multiplying this matrix together once gives the patches connected by two steps etc:
61 Summary Heterogeneity between individuals is what biology is about Extends to heterogeneity between populationsNothing is the same and doesn’t stay the sameWe haven’t touched on evolutionDispersal is universal and leads to metapopulations
62 Lecture 3. Small Populations Stochastic effectsdemographic & environmentaldemographic stochasticity in small populationsstochastic modellinge.g. death process; immigration-death processMonte Carlo simulationProbabilities of extinction
63 Stochasticity Deterministic models Demographic Environmental give expected (average) outcome (in most cases)Demographicindividuals come in single unitse.g. if deterministic model predicts 5.6 individuals, at the limit of accuracy the number can only 5 or 6Environmentalenvironments (resource availability) fluctuates “randomly”chance events (c.f. disturbance)
64 Demographic Stochasticity More predominant in small populationsrelative error is greater (c.f. plotting on log scale)Gender problemsif a population of 10 individuals produces 15 offspring, there is a 15% chance of 5 or less females
65 Pure Death Process Deterministic model Stochastic model negative exponential decay: N(t)=N(0)exp(-t)Stochastic modelexp(-t) is the probability of an individual surviving to tif individual survival is independent, then the numbers surviving to t have a binomial distribution with mean N(0)exp(-t)
66 Immigration-Death Process Solution is sum of two populationsImmigration process is a Poisson processPoisson distribution for numbers immigrating in a given timeDeath processbinomial distribution for numbers surviving to given time
68 Monte Carlo Simulation More complex models require computer simulation to find solutionsCollection of Poisson processes (i.e. assume independence between different processes)Process rates change with time
69 Monte Carlo Example Immigration-death process Process rates death: Ntotal: T = + NTime to next eventfrom negative exponential distribution with rate parameter T
70 Monte Carlo Iteration Calculate time to next event Calculate which event has occurred/T is probability of immigrationN/T is probability of deathChange populationCalculate T
71 Birth-Death Simulation Deterministic result is the mean of many stochastic resultsnot true for every modelIndividual simulations do not “look like” the meaninterpreting data
72 Random Walks Alternative view of stochastic models Population size is performing a “random walk” through timeZero is an “adsorbing barrier” (extinction)Or 1 if dioeciousAll populations (which have a death process) have a non-zero probability of reaching zero
73 Time to Extinction Death Process Birth-Death Process mean time to extinction is :Birth-Death Processfor b < d and N0 = 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
74 Probability of Extinction Assured in pure death processBirth-Death Processprob. of extinction by time t :N0 lines of descent have to become extinctprob. of ever extinction (b>d):ultimate extinction is increasingly unlikely as N0 increases
76 Effect of Increasing Rates Increasing birth rates increases variabilityIncreasing death rates decreases variabilityIn extinction probabilities, increasing variation increases extinction probabilitiesThus, for the same expected growth rate (r and R), increasing birth rates (and increasing death rates) increases chance of eventually reaching absorbing barrier
77 Logistic Results Extinction is certain Time to extinction with N0 = 1 :ln(TE) is a measure of stability of a population
79 SummaryDynamics and heterogeneity have a reciprocal effect on each otherDynamics creates variabilityVariability influences dynamicsThe mean is not always informativeThe average human being…Studying variability (and the dynamics of variability) is usually more informative than studying the mean (and the dynamics of the mean)
80 Extinction Summary Probability of extinction is less likely the greater the population sizeeach line of descent has to become extinctthe carrying capacity is largethe random walk is further from the absorbing boundarythe greater birth rates are compared to death ratesi.e. the larger the value of rthe smaller the variation in population sizethe smaller the birth rates, but see aboveExplains why the majority of populations of conservation concern tend to be large mammals in small habitats
82 Why prediction fails Models are necessarily under-specified They have to be to be usefulThe correspondence of the causal relationships they embody to actual phenomena is never known to be perfectThe observable initial conditions are never perfectly observedThere 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 unobservableParameters are estimatesSome processes are chaotic, such that arbitrarily small errors will cumulate to arbitrarily large deviations from prediction
83 Principles Exponential growth Density Dependence Circular Causality positivecooperation; aggregation; Allee effectnegativenecessary but not sufficient for stabilityCircular Causalitypathways created by interaction with environment (inc. other species)low frequency cycles / oscillationstime delaysLimiting Factorspopulations exist in complex webs of interaction, but only a few are important at particular times / places