Population Dynamics Focus on births (B) & deaths (D) B = bNt , where b = per capita rate (births per individual per time) D = dNt N = bNt – dNt = (b-d)Nt

Discrete birth intervals (Birth Pulse) Exponential Growth Density-independent growth models Discrete birth intervals (Birth Pulse) vs. Continuous breeding (Birth Flow)

 > 1  < 1  = 1 Nt = N0 t

Geometric Growth When generations do not overlap, growth can be modeled geometrically. Nt = Noλt Nt = Number of individuals at time t. No = Initial number of individuals. λ = Geometric rate of increase. t = Number of time intervals or generations.

Exponential Growth Birth Pulse Population (Geometric Growth) e.g., woodchucks (10 individuals to 20 indivuals) N0 = 10, N1 = 20 N1 =  N0 , where  = growth multiplier = finite rate of increase  > 1 increase  < 1 decrease  = 1 stable population

Exponential Growth Birth Pulse Population N2 = 40 = N1  Nt = N0 t Nt+1 = Nt 

Continuous breeding (Birth Flow) Exponential Growth Density-independent growth models Discrete birth intervals (Birth Pulse) vs. Continuous breeding (Birth Flow)

Exponential Growth Continuous population growth in an unlimited environment can be modeled exponentially. dN / dt = rN Appropriate for populations with overlapping generations. As population size (N) increases, rate of population increase (dN/dt) gets larger.

Exponential Growth For an exponentially growing population, size at any time can be calculated as: Nt = Noert Nt = Number individuals at time t. N0 = Initial number of individuals. e = Base of natural logarithms = 2.718281828459 r = Per capita rate of increase. t = Number of time intervals.

Exponential Population Growth

Exponential Population Growth

Nt = N0ert Difference Eqn Note: λ = er

Exponential growth and change over time N = N0ert dN/dt = rN Number (N) Slope (dN/dt) Time (t) Number (N) Slope = (change in N) / (change in time) = dN / dt

ON THE MEANING OF r rm - intrinsic rate of increase – unlimited resourses rmax – absolute maximal rm - also called rc = observed r > 0 r < 0 r = 0

Intrinsic Rates of Increase On average, small organisms have higher rates of per capita increase and more variable populations than large organisms.

Growth of a Whale Population Pacific gray whale (Eschrichtius robustus) divided into Western and Eastern Pacific subpopulations. Rice and Wolman estimated average annual mortality rate of 0.089 and calculated annual birth rate of 0.13. 0.13 - 0.089 = 0.041 Gray Whale population growing at 4.1% per yr.

Growth of a Whale Population Reilly et.al. used annual migration counts from 1967-1980 to obtain 2.5% growth rate. Thus from 1967-1980, pattern of growth in California gray whale population fit the exponential model: Nt = Noe0.025t

What values of λ allow What values of r allow Population Growth Stable Population Size Population Decline What values of r allow Population Growth Stable Population Size Population Decline λ > 1.0 r > 0 λ = 1.0 r = 0 λ < 1.0 r < 0

Logistic Population Growth As resources are depleted, population growth rate slows and eventually stops Sigmoid (S-shaped) population growth curve Carrying capacity (K): number of individuals of a population the environment can support Finite amount of resources can only support a finite number of individuals

Logistic Population Growth

Logistic Population Growth dN / dt = rN dN/dt = rN(1-N/K) r = per capita rate of increase When N nears K, the right side of the equation nears zero As population size increases, logistic growth rate becomes a small fraction of growth rate Highest when N=K/2 N/K = Environmental resistance

Exponential & Logistic Growth (J & S Curve)

Logistic Growth

Actual Growth

Populations Fluctuate

Limits to Population Growth Environment limits population growth by altering birth and death rates Density-dependent factors Disease, Resource competition Density-independent factors Natural disasters

Galapagos Finch Population Growth

Logistic Population Model Nt = 2, R = 0.15, K = 450 A. Discrete equation - Built in time lag = 1 - Nt+1 depends on Nt

I. Logistic Population Model B. Density Dependence

Logistic Population Model C. Assumptions No immigration or emigration No age or stage structure to influence births and deaths No genetic structure to influence births and deaths No time lags in continuous model

Logistic Population Model C. Assumptions Linear relationship of per capita growth rate and population size (linear DD) K

Logistic Population Model C. Assumptions Linear relationship of per capita growth rate and population size (linear DD) Constant carrying capacity – availability of resources is constant in time and space Reality?

I. Logistic Population Model Discrete equation Nt = 2, r = 1.9, K = 450 Damped Oscillations r <2.0

I. Logistic Population Model Discrete equation Nt = 2, r = 2.5, K = 450 Stable Limit Cycles 2.0 < r < 2.57 * K = midpoint

I. Logistic Population Model Discrete equation Nt = 2, r = 2.9, K = 450 Chaos r > 2.57 Not random change Due to DD feedback and time lag in model

Underpopulation or Allee Effect Opposite type of DD population size down and population growth down b=d b=d d Vital rate b b<d r<0 N* K N

Review of Logistic Population. Model D. Deterministic vs. Stochastic Review of Logistic Population Model D. Deterministic vs. Stochastic Models Nt = 1, r = 2, K = 100 * Parameters set deterministic behavior same

Review of Logistic Population. Model D. Deterministic vs. Stochastic Review of Logistic Population Model D. Deterministic vs. Stochastic Models Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20 * Stochastic model, r and K change at random each time step

Review of Logistic Population. Model D. Deterministic vs. Stochastic Review of Logistic Population Model D. Deterministic vs. Stochastic Models Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20 * Stochastic model

Review of Logistic Population. Model D. Deterministic vs. Stochastic Review of Logistic Population Model D. Deterministic vs. Stochastic Models Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20 * Stochastic model

Environmental Stochasticity A. Defined Unpredictable change in environment occurring in time & space Random “good” or “bad” years in terms of changes in r and/or K Random variation in environmental conditions in separate populations Catastrophes = extreme form of environmental variation such as floods, fires, droughts High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

Environmental Stochasticity A. Defined Unpredictable change in environment occurring in time & space Random “good” or “bad” years in terms of changes in r and/or K Random variation in environmental conditions in separate populations Catastrophes = extreme form of environmental variation such as floods, fires, droughts High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

Environmental Stochasticity A. Defined Unpredictable change in environment occurring in time & space Random “good” or “bad” years in terms of changes in r and/or K Random variation in environmental conditions in separate populations Catastrophes = extreme form of environmental variation such as floods, fires, droughts High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

Environmental Stochasiticity B. Examples – variable fecundity Relation Dec-Apr rainfall and number of juvenile California quail per adult (Botsford et al. 1988 in Akcakaya et al. 1999)

Environmental Stochasiticity B. Examples - variable survivorship Relation total rainfall pre-nesting and proportion of Scrub Jay nests to fledge (Woolfenden and Fitzpatrik 1984 in Akcakaya et al. 1999)

Environmental Stochasiticity B. Examples – variable rate of increase Muskox population on Nunivak Island, 1947-1964 (Akcakaya et al. 1999)

Environmental Stochasiticity - Example of random K Serengeti wildebeest data set – recovering from Rinderpest outbreak Fluctuations around K possibly related to rainfall

Exponential vs. Logistic No DD DD All populations same All populations same No Spatial component

Space Is the Final Frontier in Ecology History of ecology = largely nonspatial e.g., *competitors mixed perfectly with prey *homogeneous ecosystems with uniform distributions of resources But ecology = fundamentally spatial ecology = interaction of organisms with their [spatial] environment

Incorporating Space Metapopulation: a population of subpopulations linked by dispersal of organisms Two processes = extinction & recolonization subpopulations separated by unsuitable habitat (“oceanic island-like”) subpopulations can differ in population size & distance between

Metapopulation Model (Look familiar?) p = habitat patch (subpopulation) c = colonization e = extinction

Metapopulation Model (Look familiar?)

Rescue Effect

Another Population Model Source-sink Dynamics: grouping of multiple subpopulations, some are sinks & some are sources Source Population = births > deaths = net exporter Sink Population = births < deaths

<1 >1

Metapopulations Classic Metapopulation Definition of Population? Groups of populations within which there is a significant amount of movement of individuals via dispersal Classic Metapopulation

Metapopulation Con’t This kind of population structure applies when there are “groups” of populations occupying habitat that occurs in discrete patches (patchy). These patches are separated by areas of inhospitable habitat, but connected by routes for dispersal. Populations fluctuate independently of each other

The probability of dispersal from one patch to another depends on: Distance between patches Nature of habitat corridors linking the patches Ability of the species to disperse (vagility or mobility) – dependent on body size

Who Cares? Why bother discussing these models? Metapopulations & Source-sink Populatons highlight the importance of: habitat & landscape fragmentation connectivity between isolated populations genetic diversity