FW 453: Density-dependent population change, 2/18

FW 453: Density-dependent population change, 2/18

Overview

  • negative density dependence

  • positive density dependence

  • simultaneous negative and positive density dependence

  • simultaneous density independent and dependent factors

Negative Density Dependence

  • summary: vital rates (and the associated observed pop growth rate) decrease as the pop size increases

    • regulates pop size by decreasing growth rate at high density and increasing at low density

Case Study: Mechanism of Density Dependence in Black-throated Blue Warbler

  • as neighbords increase:

    • more time spent on territory defense and mate guarding

    • less time spent foraging

    • lower rate of nestling provisioning

    • fewer nestlings fledged

Negative Density Dependence—may affect fitness components (individual vital rates)

  • can affect more than a single component

  • flour beetle egg morality increased as the density of eggs increased

  • clutch size decreased as the density of breeding pairs increased

  • Drawn as bands because there is uncertainty and is a zone not a specific number, carrying capacity is a range.

  • Infinite number of ways in which density dependence could change vital rates and therefore per capita growth rate

Negative density dependence

  • Knowledge of the mechanism of density dependence may yield more management options

  • Possible causes of negative density dependence

    • intraspecific competition—interference/contest

    • scramble

    • disease

    • predation

    • resource availability

Population Growth

  • Geometric growth (discrete time)

    • 𝑁_(𝑡+1)=𝑁_𝑡 𝜆

  • over one time step

    • 𝜆=𝑁_(𝑡+1)/𝑁_𝑡

  • If pop grows at rate lambda for T time steps

    • 𝑁_𝑇=𝑁_0 𝜆^𝑇

    • 𝜆=√(𝑇&𝑁_𝑇/𝑁_0 )

  • Exponential growth (continuous time)

  • 𝑑𝑁∕〖𝑑𝑡=𝑟𝑁〗

  • 𝑟=ln𝜆 or 𝜆=𝑒^𝑟

  • If population grows at r for T time steps

  • 𝑁_𝑇=𝑁_0 𝑒^𝑟𝑇

Common theme

  • growth rate does NOT change at different abundances

New paradigm

  • growth rate DOES change at different abundances

The Logistic Growth Model

  • simplest way to model density dependence just adds the simplest penalty imaginable to exponential “r”.

    • dN / dt = rN

    • 𝑑𝑁∕〖𝑑𝑡=𝑟_0 𝑁(1−𝑁/𝐾)〗

      • Eq 7.3

  • Contribution to population per capita growth rate is penalized linearly; being clobbered by density same when few individuals as when a lot; penalty is the same

  • Mechanics of growth curve; sigmoidal, go through different time stages…smallish growth, highest in middle, then smallest again…maximum recruitment at K/2

Negative Density Dependence

  • “Regulates” population numbers within some equilibrium size range (carrying capacity) by:

    • decreasing population growth rate at high density and

    • increasing population growth when density is low

  • Limiting factors that determine actual equilibrium population size range

    • density-dependent

    • density-independent factors

      • have large effects on pop size

      • do not regulate population size

r vs r_0

  • r is instrinsic rate of growth in exponential/geometric growth

  • in logistic growth equation, we talk about r_0

    • the exponential growth rate at very low densities

    • similar to r_max, the theoretical maximum growth rate of a population

  • the realized per-capita growth rate is defined in Eq. 7.3

    • dN/dt = r_0N(1-(N/K))

  • the logistic growth model shows the linear decline in realized per-capita growth rate as N increases

  • logistic growth uses r_0 to show how the realized per-capita growth is affected by increasing or decreasing N

So populations stabilize at K…

  • but no always

  • lab exercise, where you use:

    • “discrete time” version of logistic, known as the ‘Ricker’ equation

    • 𝑁_(𝑡​+​1)=𝑁_𝑡 𝑒^((𝑟_0 [1−(𝑁_𝑡/𝐾)]) )

    • Note that this is exponential growth with a penalty…step through the penalty

  • IS IT DEMOGRAPHIC STOCHASTICITY? IS IT ENVIRONMENTAL STOCHASTICITY? IS IT AGE STRUCTURE?

    • No, no, and no

  • Lambda = 14.xx. Nonlinear equation with time lag….two important pieces

Deterministic chaos

  • Pop dynamics can appear chaotic when growth is very rapid

  • But the problem is in nature will not be able to determine “starting” conditions with perfect accuracy! So things will look unpredictable.

Logistic pop growth: cycles and chaos

  • Discrete logistic pop growth with high r can lead to: erratic fluctuations even in a constant environment

  • Counterintuitive finding: increasing pop growth might (temporarily) decrease pop size and increase extinction probability

Chaotic dynamics have been found in a few wild populations

  • but stochasticity will be more common in most wild pops

  • so know chaos exists but dont expect it to be common

  • Not often that little r is greater than 2.69…intrinsic growth rate…voles have exhibited chaos…most of the time not the case…usually environmental, demographic, or age structure variability

Logisitc pop growth—complexities

  • Delayed density dependence (time lags)

    • following introduction to a new range, herbivores may “irrupt” or increase to peak abundance and then crash to a carrying capacity lower than the initial peak

    • 𝑁_(𝑡+1)= 𝑁_𝑡 𝑒^(𝑟_0 [1−(𝑁_(𝑡−𝜏)/𝐾)])

  • Mechanisms:

    • effects of density on female body condition affect future reproduction

    • density affects survival cumulatively rather than in just one year

  • Nonlinear density dependence

    • (nearly) linear decline in per capita population growth (pgr) with density – as assumed by the logistic model

  • convex relationship: per capita pop. growth (pgr) varies little until near K, then drops rapidly

  • concave curve: small populations grow quickly, but pgr then declines rapidly, later flattens out – the approach to K is slow

Theta logistic pop growth model

  • Basic discrete logistic growth model

    • 𝑁_(𝑡​+​1)=𝑁_𝑡 𝑒^((𝑟_0 [1−(𝑁_𝑡/𝐾)]) )

  • Theta logistic model: the parameter theta controls the shape of the density dependence

    • 𝑁_(𝑡+1)=𝑁_𝑡 𝑒^(𝑟_0 [1−(𝑁_𝑡/𝐾)^𝜃])

  • θ ≈ 1 → Linear effects of density on population growth rate (logistic growth)

  • θ > 1 → Convex relationship (density dependence stronger at high density)

  • θ < 1 → Concave relationship (density dependence stronger at low density)

But is all density dependence negative? No!

Positive Density Dependence

  • pop growth rates increase as density increases OR

  • pop growth rates decrease as density decreases

Allee effect

  • Positive density dependence at low pop sizes: vital rates and/or pop growth rate increases as density increases

  • In other words…

    • At really low densities, pop growth can be hindered by various factors

  • Mechanisms?

    • minimize predation

      • detection and defense

      • predator confusion

      • predator swamping

    • foraging advantage

      • access to food

      • cooperative resource defense

    • reproductive success

      • finding mates

    • conditioning of environment

  • Passenger pigeon story

    • in the pigeon and dove family, but more closely related to tropical fruit pigeons than to other NA columbids

    • low reproductive rates, long-lived

    • 3-5 billion passenger pigeons when Europeans arrived

      • current pop of all birds in US is about 6 billion

    • Allee effect = human overexploitation and habitat destruction

      • chestnut blight eliminated chestnut trees and reduced food sources

      • fragmented hardwood forest habitat

    • passenger pigeons needed large flocks for courtship ritual, synchronization of mating condition

      • were never successfully captive bred

      • last one died in 1914 at age 29

  • Adding an Allee effect to our pop model; both negaitve and positive density dependence

    • basic logistic growth model

      • 𝑑𝑁/𝑑𝑡=𝑟_0 𝑁(1−𝑁/𝐾)

    • adding a threshold, A, below which per-capita growth rate becoems negative

      • 𝑑𝑁/𝑑𝑡=𝑟_0 𝑁(1−𝑁/𝐾)(1 − (𝐴+𝐶)/(𝑁+𝐶))

  • Logistic – as density goes up pgr goes down linearly;

  • Allee effect – as density increases at low densities, pgr goes up (positive dd), then becomes negative once a larger density is reached (negative dd)

  • Population decline to extinction when below A

Multiple Allee Effects

  • positive- and negative-density dependence acting on a pop

  • ex: african wild dogs

    • positive

      • increased pack size = increased likelihood of successful kill

      • increased pack size = increased prey size = more food, potentially shorter chases, and more food per individual

      • increased pack size = more eyes for protection of young, defense of food

      • increased food and defense = positive relationship between pack size and recruitment

    • Negative

      • benefit of pack size diminished at larger numbers as negative density dependence kicked in

        • too many moths to feed, etc

Including other factors

  • how does habitat quality affect carrying capacity?

    • constant over time?

  • basic idea:

    • management of habitat translates into improved conditions of some species

      • improved conditions = higher likelihood of pop persistence

    • habitat improvements can reverberate into 2 components of logisitc model

      • growth rate

      • carrying capacity (habitat carrying capacity)

  • We can incorporate habitat quality into our Ricker equation (for others)

    • must relate carrying capacity to habitat quality

      • how?

      • make CC a function of habitat quality!

    • K(X) = b_0 + b_1 * X

    • where X = a habitat covariate (e.g. rainfall, predation risk, etc)

    • can extend to many habitat factors

      • K(X_0) = b_0 + b_1 X_1 + b_2 X_2

Simple pop growth models—summary

  • Exponential: not stable

  • continuous logistic: very stable, strong tendency to go to K

  • discrete logistic: less stable, cycles, and chaos possible

  • Allee effects: very unstable. below lower equilibrium, pop can head towards extinction

Logistic pop growth model

  • Simple model whereas real pops are complex

    • interactions of density-dependent and density-independent factors

    • stochastic r, fluctuating K

    • nonlinear density dependence

    • delayed density dependence (ibex)

  • Is the model useful? It dependens…

    • simple models help us understand the general idea of density dependence

    • recruitment may be maximized at K/2

    • harvest/culling may not reduce pop size

    • fluctuations in a constant environment

  • General modeling philosophy

    • simple models are useful for describing the basics of an ecological process

    • complexity can then be added to the model, piece by piece, to represent the complexity in the real world

  • And of course, density can be applied to STAGES! ‘comonents’ vs ‘demographic DD’)

    • does the dd affect a vital rate, does that change in vital rate affect lambda, are there correlations that would cancel out the yr affect on lambda?

  • A common demographic pattern among long-lived vertebrates under negative density dependence

    • as abundance increases

      • first—juvenile survival declines (and sometimes drastically)

      • next—age of first reproduction increases

      • next—fecundity (pregnancy and fetal rate)

      • finally(and often not at all)—adult survival declines

        • changes in adult survival typically has the greatest effects on lambda

        • changes in juvenile survival usually has lesser effects on lambda

        • changes in fitness components due to negative density dependence may not always translate into effects on lambda