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

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

> 1 < 1 = 1 Nt = N 0 t

Geometric Growth When generations do not overlap, growth can be modeled geometrically. N t = N o λ t –N t = Number of individuals at time t. –N o = 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) N 0 = 10, N 1 = 20 N 1 = N 0, where = growth multiplier = finite rate of increase > 1 < 1 = 1

Exponential Growth Birth Pulse Population N 2 = 40 = N 1 N 2 = (N 0 ) = N 0 2 N t = N 0 t N t+1 = N t

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: N t = N o e rt N t = Number individuals at time t. N 0 = Initial number of individuals. e = Base of natural logarithms = r = Per capita rate of increase. t = Number of time intervals.

Exponential Population Growth

N t = N 0 e rt Difference Eqn Note: λ = e r

Exponential growth and change over time Time (t) Number (N) N = N 0 e rt Number (N) Slope (dN/dt) dN/dt = rN Slope = (change in N) / (change in time) = dN / dt

ON THE MEANING OF r r m - intrinsic rate of increase – unlimited resourses r max – absolute maximal r m - also called r c = 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 and calculated annual birth rate of 0.13.

Growth of a Whale Population Reilly et.al. used annual migration counts from to obtain 2.5% growth rate. –Thus from , pattern of growth in California gray whale population fit the exponential model: N t = N o e 0.025t

Logistic Population Growth As resources are depleted, population growth rate slows and eventually stops Sigmoid (S-shaped) population growth curve –Carrying capacity (K):

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

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 –Density-independent factors

Galapagos Finch Population Growth

Readings 8.3 through 8.13; pp Field Studies, pp Ecological Issues, p. 175 Ecological Issues, p. 219 Field Studies, pp

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

I. Logistic Population Model B. Density Dependence

I.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

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

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

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

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

I. Logistic Population Model Discrete equation N t = 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 N Vital rate N* K d b b=d b<d r<0

I.Review of Logistic Population Model D. Deterministic vs. Stochastic Models N t = 1, r = 2, K = 100

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

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

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

II.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

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

II.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)

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

II.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 All populations same DD All populations same No Spatial component

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

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

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