FW364 Ecological Problem Solving Class 11: Population Regulation October 13, 2013

Outline for Today Continue to make population growth models more realistic by adding in density dependence Objectives from Last Monday: Introduce density dependence (modeling style) Discuss scramble and contest competition Objectives for Today: Continue discussion of scramble and contest competition Introduce Allee effects (inverse density dependence) Text (optional reading): Chapter 3 Note: Midterm 2 is going to be moved to Nov. 6th, Revised syllabus available online.

Recap from Last Week How does exponential growth come to an end? Two broadly defined limits to population growth: Density independent factors Do not regulate populations Density dependent factors  Regulate populations Scramble versus Contest Exploitative, free-for-all competition Interference competition Resources shared equally Resources shared unequally Density dependence affects all individuals equally Density dependence affects some individuals lightly (territory owners) other individuals strongly (non-owners)

r (continuous) lnλ (discrete) N Contest and Scramble Comparison y-intercept: 2.00 ln λmax Contest growth rate declines faster 1.50 1.00 0.50 steady state 0.00 r (continuous) lnλ (discrete) -0.50 Carrying capacity (K) -1.00 contest Scramble growth rate does not slow down -1.50 -2.00 scramble -2.50 200 400 600 800 1000 1200 N Let’s look at λ versus population size…

Contest and Scramble Comparison 0.0 1.0 2.0 3.0 4.0 5.0 6.0 200 400 600 800 1000 1200 N y-intercept: Contest: Population growth rate declines faster at first, then slows more λmax scramble λ (discrete) contest steady state Carrying capacity (K)

Distinguishing Density Dependence Types How can we tell which type of density dependence best describes the population we are interested in? Make detailed behavioral observations OR Do a model fitting exercise i.e., Collect data on population density over time for many generations Fit the data to models of scramble and contest competition Determine which model (i.e., scramble vs. contest) fits best N λ contest scramble

Distinguishing Density Dependence Types How can we tell which type of density dependence best describes the population we are interested in? Make detailed behavioral observations OR Do a model fitting exercise N λ contest scramble Keep in mind: Are working with ideal categories… fit will not be perfect… …but categorizing is still helpful for modeling and making predictions

Population Dynamics Why do the different density dependence types matter? Scramble vs. contest has big difference on population stability i.e., change in population size through time Scramble (extreme effect) Contest Large fluctuations Increased extinction risk Difficult to predict Stable population Reduced extinction risk Easier to predict

Population Dynamics Why do the different density dependence types matter? Scramble vs. contest has big difference on population stability i.e., change in population size through time Scramble (extreme effect) Contest Large fluctuations Increased extinction risk Difficult to predict Stable population Reduced extinction risk Easier to predict Let’s investigate population dynamics more closely… … need to look at equations

General Approach to Equations Goal: Want equations for how population growth rate (λ, r) changes with density N lnλ (discrete) r (continuous) steady state Carrying capacity (K) ln λmax contest scramble Want equations to describe these functions for contest and scramble

General Approach to Equations Goal: Want equations for how population growth rate (λ, r) changes with density Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N Start with our exponential growth equations: We want to “build” equations for λ and r… Need to define new parameters: λmax and K rmax and K λmax and rmax are maximum population growth rates Occur when population size is very small (as density approaches 0, population growth rates approach maximum) K is carrying capacity  Maximum sustainable population size

Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K) Goal: Want equations for how population growth rate (λ, r) changes with density Scramble Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N (1 – Nt/K) λ = ( λmax ) (1 – N/K) r = rmax Logistic growth equations Identical to Ricker stock-recruitment equation

Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K) Goal: Want equations for how population growth rate (λ, r) changes with density Scramble Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N (1 – Nt/K) λ = ( λmax ) (1 – N/K) r = rmax Challenge: What happens to λ when: Nt is very small? Nt equals carrying capacity (K)? Nt is larger than carrying capacity (K)?

Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K) Goal: Want equations for how population growth rate (λ, r) changes with density Scramble Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N (1 – Nt/K) λ = ( λmax ) (1 – N/K) r = rmax Challenge: What happens to λ when: Nt is very small?  λ ≈ λmax (exponent ≈ 1) Nt equals carrying capacity (K)?  λ = 1 (exponent = 0) Nt is larger than carrying capacity (K)?  λ < 1 (exponent < 0)

Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K) Goal: Want equations for how population growth rate (λ, r) changes with density Scramble Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N (1 – Nt/K) λ = ( λmax ) (1 – N/K) r = rmax Challenge: What happens to r when: N is very small? N equals carrying capacity (K)? N is larger than carrying capacity (K)?

Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K) Goal: Want equations for how population growth rate (λ, r) changes with density Scramble Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N (1 – Nt/K) λ = ( λmax ) (1 – N/K) r = rmax Challenge: What happens to r when: N is very small?  r ≈ rmax N equals carrying capacity (K)?  r = 0 N is larger than carrying capacity (K)?  r < 0

Beverton-Holt equation Contest Equation Goal: Want equations for how population growth rate (λ, r) changes with density Contest Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N λmax K λ = Nt λmax – Nt + K No continuous analog Beverton-Holt equation

Contest Equation Nt+1 = Nt λ dN/dt = r N λmax K λ = Nt λmax – Nt + K Goal: Want equations for how population growth rate (λ, r) changes with density Contest Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N λmax K λ = Nt λmax – Nt + K No continuous analog Challenge: What happens to λ when: Nt is very small? Nt equals carrying capacity (K)? Nt is larger than carrying capacity (K)?

Contest Equation Nt+1 = Nt λ dN/dt = r N λmax K λ = Nt λmax – Nt + K Goal: Want equations for how population growth rate (λ, r) changes with density Contest Discrete growth: Continuous growth: Nt+1 = Nt λ dN/dt = r N λmax K λ = Nt λmax – Nt + K No continuous analog Challenge: What happens to λ when: Nt is very small?  λ ≈ λmax (denominator ≈ K) Nt equals carrying capacity (K)?  λ = 1 (denominator ≈ λmax K) Nt is larger than carrying capacity (K)?  λ < 1 (denominator > numerator)

Predicting Population Size We now have equations for population growth with density dependence! Just considering discrete growth: Scramble Contest λmax K Nt+1 = Nt Nt λmax – Nt + K (1 – Nt/K) Nt+1 = Nt ( λmax ) We can use these equations to calculate population size between any two consecutive years  We’ll do this in a graphically interesting way using replacement curves in Lab tomorrow

Replacement Curves Nt+1 Nt Nt+1 > Nt λ > 1 Plot of next year’s population (Nt+1) size against current population size (Nt) (for right now, think about points having the ability to fall anywhere in this space) Helpful for seeing how different types of density dependence influence actual population dynamics 1 : 1 line  Steady state line Nt+1 = Nt λ = 1 Nt+1 > Nt λ > 1 Nt+1 Time does not “move” from left to right in these figures!!! Nt+1 < Nt λ < 1 Example in Lab tomorrow Nt

Scramble vs. Contest Comparison Comparison of scramble and contest density dependence Similarity: Both scramble and contest density dependence (competition) result in population regulation Regulation slows population growth at high density… … which draws the population toward an equilibrium (carrying capacity) Differences: Scramble: Additional individuals always reduce the population growth rate linear relationship between lnλ (r) and N Population will overshoot K Contest: Effect of each additional individual on population growth decreases as more and more individuals are added Population will not overshoot K

Scramble vs. Contest Comparison Comparison of scramble and contest density dependence Differences Con’t: For two populations with same λmax and K 0.0 1.0 2.0 3.0 4.0 5.0 6.0 200 400 600 800 1000 1200 N Population increases faster (λ) for scramble when N < K Population decreases faster (λ) for scramble when N > K Combination drives population fluctuations for scramble (Lab 5) scramble λ contest steady state K

Another Type of Density Dependence So far, density dependence has meant: “Increased density leads to lower per capita population growth (lower λ)” The reverse can also be true in some situations! i.e., Increased density leads to higher per capita population growth (higher λ) Inverse density dependence: Allee effects Only occurs at low population sizes (at high densities, near K, regular density dependence always applies)

How Alee Effects Happen Per capita reproduction may decrease at low population sizes  Mate limitation: density gets so low that mates cannot find each other  Inbreeding: density so low that inbreeding occurs  For plants, low density may mean fewer pollinators attracted Mate limitation Inbreeding Pollinator attraction Per capita mortality may decrease at higher population sizes (i.e., increase at low size)  “Safety in numbers” situation, e.g. colony nesting  For plants, greater density may mean greater soil stability (less erosion) Colony nesting Soil erosion

How Alee Effects Happen Constant harvest amount also functions like an Allee effect Lab 3: Nt+1 = Nt (1 + b’ – d’) - H When a constant number is harvested, the proportion of the population harvested changes as population size changes For H = 100 When N = 1000, the proportion harvested is 0.10 ( = 100 / 1000) When N = 400, the proportion harvested is 0.25 ( = 100 / 400) When N = 200, the proportion harvested is 0.50 ( = 100 / 200) Proportional harvest mortality increases as population decreases (like a death rate increasing at low population sizes) … inverse density dependence  Constant harvests are a risky practice Blue whale harvest

Alee Effects Dynamics Allee effects involve positive feedback at low population sizes i.e., population decline leads to further decline Result: Dip in the population growth rate versus density curve λ > 1 scramble without Allee effect scramble with Allee effect Allee effect gets stronger from top to bottom curve λ < 1 Based on Fig. 3.13

Alee Effects Dynamics Allee effects involve positive feedback at low population sizes i.e., population decline leads to further decline Result: Dip in the population growth rate versus density curve λ > 1 scramble without Allee effect has one equilibrium = K scramble with Allee effect has two equilibria λ < 1 Based on Fig. 3.13

Allee Effects – Population Forecast Allee Replacement Curve Stable equilibrium Unstable equilibrium N0 below first equilibrium  Population crash N0 above first equilibrium  Carrying capacity

Allee Effects – Population Forecast Allee Replacement Curve Stable equilibrium Unstable equilibrium Fate of population depends on starting conditions (starting size)

Allee Effects – Population Forecast Allee Replacement Curve λ > 1 Increase to K Decrease to K Decline to extinction λ < 1 Population fate is extinction if it falls below unstable equilibrium

Allee Effects – Wrap Up Allee effects have major implications for management of endangered species and fisheries  It is crucial that Allee effects be identified and quantified  Must manage to keep population far above “the well” (unstable equilibrium) Possible that Allee effects are the cause of collapses of some overharvested fisheries Bluefin tuna Even if population does not decline at low densities, a slowed rate of growth will keep the population low for longer periods (slow to recover)  More vulnerable to random extinction

Lab Tomorrow We will be working with some data in excel and doing some practice problems similar to what you will be seeing on Midterm 2 (now scheduled for Nov. 6th)  We will look at scramble and contest competition  We will look at Allee effects You won’t need to bring the RAMAS software, as we will be working with excel and mainly analyzing data.