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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. Immigration-Death Solution

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

  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 equilibrium

  10. Immigration-death model with different L

  11. 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

  12. 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

  13. 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

  14. 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)

  15. 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

  16. 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

  17. Logistic Equation - Empirical • Empirical observations combined • Fits many, many data

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

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

  20. Logistic Equation - Dynamics

  21. Logistic Equation Properties

  22. 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

  23. 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

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

  25. 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

  26. 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

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

  28. 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

  29. 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

  30. 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…

  31. Outcomes

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

  33. 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

  34. 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)

  35. 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

  36. 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

  37. Lecture 2: Structuring Populations • Age • Leslie matrices • Metapopulations • Probability distributions • Metapopulations • Levin’s model

  38. 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

  39. 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

  40. 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

  41. 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

  42. Leslie Matrix • This difference equation can be written in matrix notation

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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)

  48. 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)