458 Generation-Generation Models (Stock-Recruitment Models) Fish 458, Lecture 20

458 Recruitment Annual recruitment is defined as the number of animals “added to the population” each year. However, recruitment is also defined by when recruitment occurs : at birth (mammals and birds); at age one (mammals and birds, some fish); at settlement (invertebrates / coral reef fishes); when it is first possible to detect animals using sampling gear; and when the animals enter the fishery. All of these definitions are “correct” but you need to be aware which one is being used.

458 Stock and Recruitment - Generically (the single parental cohort case) The generic equation for the relationship between recruitment and parental stock size (spawner biomass in fishes) is: Recruitment equals parental numbers multiplied by survival, fecundity and environmental variation. The functional forms allow for density- dependence.

458 Stock and Recruitment - Generically (the single parental cohort case) Consider a model with no density- dependence: The population either grows forever (at an exponential rate) or declines asymptotically to extinction. The must be some form of density- dependence!

458 Some Hypotheses for Density-Dependence Habitat: Some habitats lead to higher survival of offspring than others (predators / food). Selection of habitat may be systematic (nest selection) or random (location of settling individuals). Fecundity Animals are territorial – the total fecundity depends on getting a territory. Feeding Given a fixed amount of food, sharing of food amongst spawners will occur.

458 A Numerical Example-I Assume we have an area with 1000 settlement (or breeding) sites. Only one animal can settle on (breed at) each site. The factors that impact the relationship between the number attempting to settle (breed) and the number surviving (breeding) depends on several factors.

458 A Numerical Example-II Hypothesis factors: Sites are selected randomly / to maximise survival (breeding success). Survival differs among sites (from 1 to 0.01) or is constant. Attempts by more than one animal to settle on a given site leads to: finding another site (if one is available), death (failure to breed) for all but one animal, death of all the animals concerned. How many more can you think of??

458 Case 1: No density-dependence (below 1000) Survival is independent of site; individuals always choose unoccupied sites (or they choose randomly until they find a free site).

458 Case 2 : Site-dependent survival (optimal site selection) Survival depends on site; individuals always choose the unoccupied site with the highest expected survival rate.

458 Case 3 : Site-dependent survival (random site selection) Survival depends on site; individuals choose sites randomly until an unoccupied site is found.

458 Case 4 : Site-independent survival (random site selection) Survival is independent of site; individuals choose sites randomly but die / fail to breed if a occupied site is chosen.

458 Case 5 : Site-independent survival (competition among occupiers). Survival is independent of site; individuals choose sites randomly but if two (or more) individuals choose the same site they all die / fail to breed.

458 Numerical Example (Overview of results) Depending on the hypothesis for density-dependence: Recuitment may asymptote. Recruitment may have a maximum and then decline to zero. We shall now formalize these concepts and provide methods to fit stock- recruitment models to data sets.

458 Selecting and Fitting Stock- Recruitment Relationships

458 The Beverton-Holt Relationship The survival rate of a cohort depends on the size of the cohort, i.e.: This can be integrated to give:

458 The Ricker Relationship The survival rate of a cohort depends only on the initial abundance of the cohort, i.e: This can be integrated to give:

458 A More General Relationship The Ricker and Beverton-Holt relationships can be generalized (even though most stock- recruitment data sets contain very little information about the shape of the stock- recruitment relationship):

458 The Many Shapes of the Generalized Curve

458 Fitting to the Skeena data We first have to select a likelihood function to fit the two stock-recruitment relationships. We choose log-normal (again) because recruitment cannot be negative and arguably whether recruitment is low, medium or high (given the spawner biomass) is the product of a large number of independent factors.

458 The fits ! Negative log-likelihood Beverton-Holt: Ricker:

458 Readings Burgman et al. (1993); Chapter 3. Hilborn and Walters (1992); Chapter 7. Quinn and Deriso (1999); Chapter 3.