1 / 33

FW364 Ecological Problem Solving

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)

kyoko
Download Presentation

FW364 Ecological Problem Solving

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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.

  3. 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 Contest Scramble versus Exploitative, free-for-all competition Interference competition Resources shared equally Resources shared unequally Density dependence affects some individuals lightly (territory owners) other individuals strongly (non-owners) Density dependence affects all individuals equally

  4. Contest and Scramble Comparison y-intercept: 2.00 lnλmax Contest growth rate declines faster 1.50 1.00 0.50 steady state 0.00 lnλ(discrete) r (continuous) -0.50 Carrying capacity (K) contest -1.00 Scramble growth rate does not slow down -1.50 -2.00 scramble -2.50 0 200 400 600 800 1000 1200 N • Let’s look at λ versus population size…

  5. Contest and Scramble Comparison 6.0 y-intercept: • Contest: Population growth rate declines faster at first, then slows more λmax 5.0 4.0 scramble λ(discrete) 3.0 2.0 contest steady state 1.0 Carrying capacity (K) 0.0 0 200 400 600 800 1000 1200 N

  6. 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 scramble λ contest N

  7. 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 scramble λ • Keep in mind: • Are working with ideal categories… fit will not be perfect… • …but categorizing is still helpful for modeling and making predictions contest N

  8. 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

  9. 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

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

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

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

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

  14. Scramble Equation • Goal: Want equations for how population growth rate (λ, r) changes with density • Discrete growth: • Continuous growth: • dN/dt = r N • Nt+1 = Ntλ • (1 – Nt/K) • (1 – N/K) • λ =( λmax ) • 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

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

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

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

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

  19. Contest Equation • Goal: Want equations for how population growth rate (λ, r) changes with density • Discrete growth: • Continuous growth: • dN/dt = r N • Nt+1 = Ntλ • No continuous analog • λmax K • λ = • Ntλmax – Nt + K • 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) • Contest

  20. Predicting Population Size • We now have equations for population growth with density dependence! • Just considering discrete growth: • λmax K • (1 – Nt/K) • Nt+1 = Nt • Nt+1 =Nt ( λmax ) Contest Scramble • Ntλmax – Nt + K • 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

  21. Replacement Curves • Replacement Curves: • 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 Nt+1>Nt λ > 1 1 : 1 line  Steady state line Nt+1 Nt+1 = Ntλ = 1 • Time does not “move” from left to right in these figures!!! • Example in Lab tomorrow Nt+1 < Nt λ < 1 Nt

  22. 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 overshootK • Contest: Effect of each additional individual on population growth decreases as more and more individuals are added • Population will not overshootK

  23. Scramble vs. Contest Comparison • Comparison of scramble and contest density dependence 6.0 • Differences Con’t: • For two populations with same λmaxand K 5.0 4.0 • Population decreases faster (λ) for scramble when N > K • Population increases faster (λ) for scramble when N < K 3.0 2.0 scramble λ 1.0 • Combination drives population fluctuations for scramble (Lab 5) contest steady state 0.0 K 0 200 400 600 800 1000 1200 N

  24. 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)

  25. 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

  26. 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) • Blue whale harvest • 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

  27. 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

  28. 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

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

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

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

  32. Allee Effects – Wrap Up • Allee effects have major implications for management of endangered species and fisheries •  It is crucial that Alleeeffects 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

  33. 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.

More Related