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