1. Quantitative Biology: populations

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

3. Definitions Population Metapopulation Community
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
Compare 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 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
dN/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 rate
There 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 equilibrium
For 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.

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
The 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 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 Limiting 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
The 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 population
Density dependence
Birth and death are universal for biological populations
The direct implication is that populations are regulated

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

26. 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

27. 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

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

29. 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

30. 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

31. 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…

32. (initial condition dependent)
Outcomes
K2 > K1/12
Spp 2 is more competitive at high densities
K2 < K1/12
Spp 2 is less competitive at high densities
K1 > K2/21
Spp 1 is more competitive at high densities
Exclusion
(initial condition dependent)
Spp 1 wins
K1 < K2/21
Spp 1 is less competitive at high densities
Spp 2 wins
Co-existence

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

35. 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

36. 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)

37. 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

38. 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

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

40. 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

41. 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

42. 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

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 Matrix
This difference equation can be written in matrix notation

45. 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

46. Leslie Matrix Example
This 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 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

48. 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

49. 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)

50. 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)

51. 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

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 p
Dispersal occurs all the time, tending to increase small populations

54. Model with Decreasing Extinction
Include dependence of local extinction on total patch occupancy
Exponential 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 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 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 What is
the Allee effect? Oikos. 87, ).

58. Patches as “Networks”
Simplest models have all patches equally connected
But 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 populations
Nothing is the same and doesn’t stay the same
We haven’t touched on evolution
Dispersal is universal and leads to metapopulations

62. 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

63. Stochasticity Deterministic models Demographic Environmental
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)

64. 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

65. Pure Death Process Deterministic model Stochastic 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)

66. 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

67. Stochastic Population Processes

68. 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

69. Monte Carlo Example Immigration-death process Process rates
death: N
total: T =  + N
Time to next event
from 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 immigration
N/T is probability of death
Change population
Calculate T

71. 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

72. 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

73. Time to Extinction Death Process Birth-Death Process
mean time to extinction is :
Birth-Death Process
for 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 process
Birth-Death Process
prob. of extinction by time t :
N0 lines of descent have to become extinct
prob. of ever extinction (b>d):
ultimate extinction is increasingly unlikely as N0 increases

76. 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

77. Logistic Results Extinction is certain Time to extinction
with N0 = 1 :
ln(TE) is a measure of stability of a population

79. 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)

80. 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

81. Lecture 4. Modelling

82. 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

83. Principles Exponential growth Density Dependence Circular Causality
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