Quiz

Depensation and low density dynamics

References Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. Canadian Journal of Fisheries and Aquatic Sciences 54:1976-1984 Liermann M & Hilborn R (2001) Depensation: evidence, models and implications. Fish and Fisheries 2:33-58 Myers RA, Barrowman NJ, Hutchings JA & Rosenberg AA (1995) Population dynamics of exploited fish stocks at low population levels. Science 269:1106-1108

Definitions Compensatory: rate of increase declines at higher densities Depensatory: rate of increase declines at lower densities

Why low density This is where the conservation concern is!

What happens at low densities Competition for resources should be at a minimum BUT Reduced fertilization due to difficulty in finding mates All births one gender Random events with small numbers Reduced survival due to predation Genetic inbreeding depression Lack of benefits from grouping behavior

The probability of finding a mate Mate finding The probability of finding a mate Assume that the probability a female will encounter any individual male is p and there are N males in the population. Expected average number of males encountered is λ = pN The total number of encounters with males is Poisson distributed The probability that no males will be encountered will be the 0 class of the Poisson distribution with mean = λ.

Poisson: probability of 0 class Mate finding Poisson: probability of 0 class

Mate finding Proportion not mated and mated e.g. p = 0.0138 probability of finding a male 12 Depensation and extinction I.xlsx

(the population size at which 50% are mated) Mate finding Reparameterizing to N50 (the population size at which 50% are mated)

Mate finding N50 10 20 50 100 200 p 0.0693 0.0347 0.0139 0.0069 0.0035 12 Depensation and extinction I.xlsx

Single-gender All births one gender If there are N total births in a population and the proportion of females is p = 0.5 What is the probability that zero births are female, or all births are female? This is a binomial distribution, probability of 0 females out of N births OR N females out of N births

Single-gender cohorts 12 Depensation and extinction I.xlsx

Random events with small numbers Randomness Random events with small numbers Population in equilibrium, births = deaths Each time step for each individual, probability b of producing one offspring, probability d of dying, probability 1 – b – d of neither dying nor reproducing Set b = d, simulate for 100 years with different starting population sizes How often does the population go extinct? This is an individual-based model, called a “random walk” model

R code: Random walk model.r Randomness R code: Random walk model.r

Randomness 5 individual population simulations (starting population size = 20, b = d = 0.2, 100 years) 12 Depensation and extinction I.xlsx 12 Random walk.r

Probability of extinction before year 100 Randomness Probability of extinction before year 100 (100 simulations, 100 years, 100 individuals = 1 million calculations, 50 seconds) 12 Depensation and extinction I.xlsx 12 Random walk.r

Probability of extinction before year 100 Randomness Probability of extinction before year 100 (1000 simulations, 100 years, 100 individuals = 10 million calculations, 15 minutes) 12 Depensation and extinction I.xlsx 12 Random walk.r

Probability of extinction before year 100 Randomness Probability of extinction before year 100 (10000 simulations, 100 years, 100 individuals = 100 million calculations, 1 hr 23 min) 12 Depensation and extinction I.xlsx 12 Random walk.r

Predation at low densities Predator might be very efficient Predator population unaffected by prey abundance (i.e. this prey species is not the major part of the diet Then predator functional response can lead to depensatory mortality (higher mortality per capita at low densities)

Predation

Functional response a = 300, b = 400 a = 300 b = 400 a = 300 b Predation a = 300, b = 400 Functional response a = 300 Max prey eaten b = 400 N when prey eaten is at half of max a = 300 b 12 Depensation and extinction I.xlsx

Depensation due to predation Depensation: predation Depensation due to predation Add predation to stock-recruit relation (salmon) R = Rstock-recruit – number eaten Functional response leads to depensation (higher mortality per capita at low densities) Rate of increase = recruits divided by spawners (salmon) 12 Depensation and extinction I.xlsx

Equilibrium and critical depensation Depensation: predation Equilibrium and critical depensation Unfished equilibria Critical depensation 12 Depensation and extinction I.xlsx

Depensation due to mating success Depensation: mating Depensation due to mating success Replace spawners S in stock-recruit with pmated × S Proportion mated Number of spawners at which 50% successfully mate A Beverton-Holt curve 12 Depensation and extinction I.xlsx

Low densities (summary) Increased risk of extinction All births one gender Random events Predation Difficult to find mates Other (inbreeding, lost group benefits, etc.) The net effect is depensation: lower rate of increase at low densities

Ransom Myers 1952-2007 Myers analysis Myers RA, Barrowman NJ, Hutchings JA & Rosenberg AA (1995) Population dynamics of exploited fish stocks at low population levels. Science 269:1106-1108

Detecting depensation Myers analysis Detecting depensation Model 1: δ = 1 (find MLE, likelihood L1) Model 2: δ free (find MLE, likelihood L2) Nested model Likelihood ratio test: R = 2ln(L2/L1) is a chi-square distribution with degrees of freedom 1

Compare model 1 and model 2 Myers analysis delta=1 delta free alpha 0.87 0.07 K 3330.0 7116.9 delta 1 1.78 sigma 1.01 0.66 NLL 32.79 23.12 nparams 3 4 Likelihood ratio 19.35 Degrees of freedom Chi-squared prob 1.1E-05 Compare model 1 and model 2 12 Depensation Myers.xlsx

Myers results Explored 128 data sets Myers analysis Myers results Explored 128 data sets Only 3 significant cases of depensation Fewer than expected by chance Of these data sets about 27 had high power

Problems with Myers method Myers analysis Problems with Myers method Parameterization has no biological interpretation except δ > 1 implies depensation Used p values to test for significant depensation, ignores biological significance Confounding of environmental change (regime shifts) with depensation