Population Models• So far we’ve focused on:– Estimation techniques– Characteristics of populations• Introduction to population growth models– Unlimited resources, density indpendent growth– Limited resources, density dependent growthPopulation Models• Forecast future conditions– Sustainable yield – Population viability analysis– Trajectory of population size for invasive species• Hindcast to explore potential mechanismsPopulation GrowthExamplesHuman population American shad, Columbia River1940 1950 1960 1970 1980 1990 2000 201Annual Dam Count at Bonneville 0100000020000003000000400000050000006000000Population Growth: a simple case• Constant environment• Unlimited resources• All animals are the samePopulation Growth: a simple case• Change in numbers = ∆N • ∆N = (Births – Deaths) + (Immigrants-Emigrants)• Ignore immigrants and emigrants – Assume closed population or– assume I = E– or combine B + I and D + E• ∆N = B - DPopulation Growth: a simple case• ∆N = N1–N0•N1–N0= Births – Deaths•N1–N0= RBN0–RDN0 Population Growth: a simple case• ∆N = N1–N0•N1–N0= Births – Deaths•N1–N0= RBN0–RDN0 •N1= N0+ RBN0–RDN0 •N1= N0( 1 + RB–RD)Population Growth: a simple case• ∆N = N1–N0•N1–N0= Births – Deaths•N1–N0= RBN0–RDN0 •N1= N0+ RBN0–RDN0 •N1= N0( 1 + RB–RD)• λ = ( 1 + RB–RD)•N1= N0λPopulation Growth: a simple case• ∆N = N1–N0•N1–N0= Births – Deaths•N1–N0= RBN0–RDN0 •N1= N0+ RBN0–RDN0 •N1= N0( 1 + RB–RD)• λ = ( 1 + RB–RD)•N1= N0λ• Note: λ = N1/ N0Population Growth: a simple case• Let’s project:•N2= N1λPopulation Growth: a simple case• Let’s project:•N2= N1λ•N2 =(N0λ) λ•N2 = N0λ2Population Growth: a simple case• Let’s project:•N2= N1λ•N2 =(N0λ) λ•N2 = N0λ2•N3= N0λλλ•N4= N0λλλλ, etc…• In general, Nt= N0λt• Nt= N0λtNtPopulation Growth: a simple case• This treatment of growth rate (λ) is very simple and intuitive: λ = 1.06 = 6% increase per yearAssumptions• Birth rate is constant• Death rate is constant• It treats all members of the population as equal or it assumes a stable age distribution– Reasonable for some populations• Non-overlapping generations (insects, annual plants)• λ can also be estimated from age-specific birth and death rates • Note: only describes the population size per time interval– What if species does not have seasonal reproduction?– What if we want compare species with different intervals of population change (the tortoise vs. the hare)– λ is mathematically cumbersome in some instances. Use calculus-based continuous-time analogdN/dt = r N• Nt= N0λt• ln (Nt) = ln (N0) + (ln λ) t• ln (Nt) = ln (N0) + r t• y = a + b x• Nt= N0λt• ln (Nt) = ln (N0) + (ln λ) t• ln (Nt) = ln (N0) + r t• y = a + b xln (N)tNt• Nt= N0λt• ln (Nt) = ln (N0) + (ln λ) t• ln (Nt) = ln (N0) + r t• y = a + b xln (N)tr• r = ln (λ) • λ = errlambda-0.51083 0.6-0.22314 0.8-0.05129 0.95010.04879 1.050.182322 1.20.336472 1.40.69314722.30258510ln (N)tFinite and Instantaneous Ratesλ = 1.13 / year What is the daily rate?Finite and Instantaneous Ratesλ = 1.13 / year What is the daily rate?• Instantaneous rates can easily be subdivided, but finite rates can’t:λ = 1.13 / 12 = 0.094 / month = 90.6% decrease / monthFinite and Instantaneous Rates• Instantaneous rates can easily be subdivided, but finite rates can’t:λ = 1.13 / year r = 0.12 / year = 0.01 / month = 0.000329 / dayλ = er= 1.000329 / day = 0.03% /dayFinite and Instantaneous Rates• Finite survival rate:S = Nt/ N0• Instantaneous mortality rate:z = - ln (S)S = e-zFinite and Instantaneous Rates• Finite survival rates are multiplicative• Instantaneous mortality rates are additive:zweek= 7 zdailyUnlimited Growth Assumptions• b – d is constant, implies constant environment and unlimited resources• All members of the population are equal or population has a stable age distribution– Reasonable for some populations• Non-overlapping generations (insects, annual plants)• Nonetheless, simple exponential growth models provide good predictions in many cases, e.g., collared dove in EnglandChanging environment• Population change is in the real world is dynamic, b and d change• Observed change is caused by:– Real changes (Process error)• deterministic causes• stochastic process error– factors we don’t know about– true randomness, e.g. demographic stochasticity– Sampling (observation) error • We can incorporate into modelsStochastic population growth• Mills 2007 Figure 5.5Stochastic population growtht lambda N010011.212021.214431.217341.220751.2 24961.229971.235881.2 430Stochastic population growtht lambda N Lambda N0100-10011.21201.2 1202 1.2 144 1.4 1683 1.2 173 1 1684 1.2 207 1.1 1855 1.2 249 1.3 24061.22991.1 26471.23581.3 34481.24301.2 412aver age 1.2 1.2St d D e v 0 . 0 000.131Stochastic population growtht lambda N Lambda N Lambda N0 100 - 100 - 1001 1.2 120 1.2 120 1.2 1202 1.2 144 1.4 168 1.6 1923 1.2 173 1 168 0.8 1544 1.2 207 1.1 185 0.9 1385 1.2 249 1.3 240 1.5 2076 1.2 299 1.1 264 0.9 1877 1.2 358 1.3 344 1.5 2808 1.2 430 1.2 412 1.2 336average 1.2 1.2 1.2Std Dev 0.000 0.131 0.312Stochastic population growtht lambda N Lambda N Lambda N0 100 - 100 - 1001 1.2 120 1.2 120 1.2 1202 1.2 144 1.4 168 1.6 1923 1.2 173 1 168 0.8 1544 1.2 207 1.1 185 0.9 1385 1.2 249 1.3 240 1.5 2076 1.2 299 1.1 264 0.9 1877 1.2 358 1.3 344 1.5 2808 1.2 430 1.2 412 1.2 336Arit hmet ic Mean 1.2 1.2 1.2Std Dev 0.000 0.131 0.312Geometric Mean 1.200 1.194 1.164λG= (λ1* λ2* λ3*…* λt)1/tUnlimited growth summary• Unrealistic (long-term) assumptions• Finite and instantaneous forms each have advantages• Stochasticity affects ability to accurately predict future conditions• Provides accurate predictions in many cases– short time intervals– invading/colonizing populations–post-disturbance