Pedro F. Quintana- Ascencio, Office: Biology Bldg. 401 E Phone: / Office hours : Tuesday & Thursday: 9:00-11:00 Population Viability Analysis

What is population viability analysis? The use of quantitative methods to predict the likely future status of a population or collection of populations of conservation concern

Population viability analysis Provide assessments of population persistence based on a combination of empirical data and modeling scenarios

Advantages of modeling Analyze and synthesize data Make assumptions and data limitations transparent Formulate the logic of the research problems in a consistent and unambiguous way

Potential products and uses of PVA Assessing the extinction risk of populations Anticipate demographic changes Compare relative risks of several populations Assessment of Risk

Potential products and uses of PVA Identify key life stages or demographic processes as management targets Determine how large a population needs to be to avoid extinction Determine how many individuals to release Setting limits to harvesting compatible with persistence Deciding how many populations to protect Guiding management

Other uses and products? Spatial partitioning of individuals among populations Genetic partition of individuals Role of corridors Spatial architecture of networks Multiple species approach (community viability?) Environmental contextual dynamics Assessing social structure

How to describe demographic change ? Number of individuals Population structure Individual characteristics Population number and structure

Types of PVAs Count-based Data needed: exhaustive counts or estimates of the total number of individuals, or of a subset of individuals in the population Assumptions: All individuals are identical

Types of PVAs Demographic models Data needed: rates of demographic processes separately for each type of individual in the population Limitations: require more and data

Types of PVAs Multi site models Data needed: measures of local occupancy, and estimates of rates of movement between populations and of rates of local extinction. Information on spatial structure and environmental correlation Limitations: They may require extensive data

Types of PVAs Individual based models Data needed: information on actual local and behavior of all individuals Limitations: They require the most detailed data

Quantitative Conservation Biology Williams F. Morris y Daniel F. Doak (2002). Sinauer Associates

A model philosophy Morris y Doak recommend: “Keep it simple” “Let the available data tell you which type of PVA to perform” “Make sure you know what your model is doing”

Tinkering !!!!

Authors urge you to: Understand the structure of their programs as a way to truly know how the underling models work Combine and modify them to suit your needs

Mark Shaffer and collaborators Their seminal work used computer simulations to evaluate if the population of “grizzly bear” (Ursus arctos) in the Greater Yellowstone ecosystem had at least a 0.95 chance of surviving for different periods in the future.

The Grizzly Bear 100,000 were estimated in 1800 Today there are only 1000 They estimated that local populations needed 100 bears to have a 95 % chance of survival

Their results indicated a high probability of persistence for the 100 years but a more uncertain future during the next 300 years. Their conclusions affected the management within the park and had legal consequences.

Russell Lande and collaborators Studied changes in the populations of the Northern Spotted Owl (Strix occidentalis) in the context of the logging of old growth forests on which the owl depends.

Their results suggest that populations may be declining but the results were not enough to eliminate other scenarios including stable populations.

Deborah Crouse and collaborators They evaluated the relative effect of two human activities on the persistence of the loggerhead sea turtle (Caretta caretta) in the southeastern coast of the United States: trampling of eggs and hatchlings on beaches and drowning of older-aged turtles in fishing nets were hypothesized to underline their declining numbers. Crouse et al. 1987

Their results showed that the use of mechanisms that reduce the mortality of adult turtles is much more effective to increase these populations.

The African elephant Lande and collaborators analyzed how large should be the national paks in Africa to maintain viable populations of elephants (Loxodonta africana) Their conclusions suggest that at least 2500 km 2 are necessary

An example:

An example: Population change Change in population size during time interval Births during time interval Deads during time interval = _ N t = B _ D N t+1 = N t + B - D

Per capita birth rate Is the number of offspring produced per unit time by an average member of the population If there are 34 births per year in a population of 1000 the annual per capita birth rate is 34/1000=0.034

Use of the rates b=0.034, N=500 B=bN B=0.034*500 B= 17 per year

Instantaneous rate of population size change N t = B _ D N t = bN _ mN = b _ m r N t = rN d N d t = rN

This equation can be solved by integration to give the familiar equation for exponential population growth N t = N o e (b-d)t N t = N o e rt e = the base of the natural logarithms r = intrinsic rate of natural increase

We can replace e r with λ N t = N o λ t λ = finite rate of increase

The exponential model describes population growth in a idealized, unlimited environment

Useful terminology Let A x B be a product set consisting of all ordered pairs (a, b) where a is a member of A and b is a member of B. Then any subset of A x B is called a relation

Useful terminology Function: is a relation that can uniquely identify an element in the domain for each element in the range Domain: The set A of the variable is called the domain Range: The set B is defined in such a way that all members of B are associated with members of A

Model Is an equation describing the relationship between the independent variables and the dependent ones

Dependent variable: the thing in the model you want to estimate this entity depend on other factors Independent variable: these other factors Parameters: those components that mediate the relationship between independent and dependent variables Useful terminology

Then?? N t = N o λ t λ = finite rate of increase

Model assumptions There are no density dependent effects Births and Deaths are mutually independent B and D are also independent of the age of the individuals B and D are constant in time There is no uncertainty in the prediction

Parameters and initial conditions 57 rhinoceros (45 adults, 4 yearlings + 8 juveniles) Another assumption: Females are usually the limiting sex in reproduction Birth rate =0.14 per year Death rate= 0.08 per year r= ( ) = 0.06 In 1986 there were 35 females Please predict the number the individuals under the assumption of 1:1 ratio after 50 years

Conway and Goodman observed a 50 % death rate in the transition from yearling to juvenile It is not strictly important that rates are different in different age classes if the proportion of the population within each age class remains more or less constant We will demonstrate this later By now we should assume this so

For the purpose of the management plan we will evaluate the next 50 years Our starting date will be 1986 The deterministic prediction: N t = 35e 0.06 * t N t = 35* t

Deterministic prediction

Adding realism: using integer numbers Algorithm For each time step from1 to t, do steps 2 to 7 2.Let N(t+1) take the value of the current pop size N(t) 3.For each animal from 1 to N(t+1), do steps 4 to 7 4.Choose a uniform random number U1 5.Choose a uniform random number U2 6.If U1 is less than d then decrease N(t+1) by 1 7.If U2 is less than b, then increase N(t+1) by 1

Carrying capacity (K) Maximum population size that a particular environment can support

The logistic Growth Model d N d t = r max N (K-N) (K)

d N d t = r max N (K-N) (K) K-N = the additional number of individuals that the environment can accommodate (K-N)/K = the proportion of K that is still available for population growth