Spatial Structure and Metapopulations “Metapopulation biology represents one way of explicitly putting population biology into a spatial context.” Hanski and Gaggiotti (2004). So far the models that we have considered assume that the population is closed (no immigration or emigration). However, this assumption is probably not valid for many species. Movement of individuals among populations is common and can have important theoretical and management implications. We will discuss a class of population models that take into account the movement of individuals among populations or habitats and how such movement might influence the persistence of a population. Such models are generally referred to as metapopulation models, with a metapopulation consisting of several populations that are linked by migration and rely on the existence of several features: 1. separate breeding populations in discrete habitat patches 2. the absence of any population large enough to sustain all others 3. sufficient isolation to make dispersal between habitat patches rare 4. Sufficiently asynchronous local dynamics so that populations at all habitat patches do not go extinct at once. (M. Alex Smith, http://www.redpath-staff.mcgill.ca/asmith/PROJECT3.HTM) Metapopulation theory implicitly or explicitly includes spatial attributes and metapopulation biology is considered a subdiscipline of spatial ecology We will discuss two types of metapopulation models. 1. One type of model is not concerned with the size or density of any particular population (i.e. N or N/A) or with the dynamics within a population. These models are typically concerned with the persistence (extinction and colonization) of particular habitat areas. 2. The other model that we will discuss is the source-sink model developed by Pulliam (1988). This model is a more traditional BIDE model and takes into consideration the internal dynamics of particular habitats and links them by immigration and emigration. The Levins Model We will develop a series of models, using the notation and derivation found in Gotelli (1998). One of these models was developed by Levins (1969) and the other models are modifications of the Levins’ model. A caution: Gotelli uses different notation from Levins.Let f= the fraction of sites occupied and the change in f over time is determined by losses due to extinction (E) and gains from colonization or immigration (I) EIdtdf−= Model 1: Let pi=the probability of local colonization Immigration (I) depends on the pi and on the fraction of sites that are unoccupied (1-f) and available to colonize Therefore ()fpIi−= 1 Extinction (E) depends on the fraction of sites occupied (f) and the probability of local extinction, which equal pe Therefore, ()fpEe= Substituting gives ()(fpfpd)tdfei−−= 1 if we initially assume that pi and pe are constant, then we can set this equation equal to zero and solve for f eiipppf+=^ Model 2: This is Gotelli’s version of the model proposed by Levins We relax the assumption of constant pi and let pi be a function of the fraction of sites occupied.ifpi= i is a measure (a constant) of much the probability of colonization increases with each additional patch that is occupied. Substituting gives ()(fpfifd)tdfe−−= 1 again we can set this equal to zero and solve for f ipfe−=1^ Levins equation looks like this eppmpdtdf−−= )1( Set this equal to zero and solve mep −=1 Hanski (1991) extends the model to include compensatory effects by making extinction and colonization explicit functions of isolation and area, using negative exponential functions. ()aDmm −= exp0 ()bAee −= exp0 The equilibrium value of p becomes ()aDbAmep +−⎟⎟⎠⎞⎜⎜⎝⎛−= exp1ˆ00The source/sink model (this is a summary of the model developed in Pulliam 1988) The source sink model was developed to describe patterns of habitat selection ***. We are going to assume that BIDE are deterministic but that they vary among habitats. Let total births equal the sum of births in all habitats ∑==mjjbB1 Similarly total deaths equal the sum of deaths in all habitats ∑==mjjdD1 Movement among habitats consists of ijk= immigration from habitat k into habitat j ∑∑===+=mkjkjmkjkjiiii001 ejk= emigration from habitat j into habitat k ∑∑===+=mkjkjmkjkjeeee001 If we consider all habitats then I and E are negligible (i.e. movement into or out of the entire system). ∑∑======mjjmjjeEiI10100 If the system is in equilibrium then B=D and λ=1 However, λ doesn’t necessarily equal 1 in each habitatSources are defined as habitats in which bj > dj and ej > ij and sinks are defined as habitats in which bj < dj and ej < ij Population Regulation Consider the following population model 1. there are two habitats 2. one habitat is a source and by definition λ1>1 3. one habitat is a sink and by definition λ1<1 4. The limiting resource in habitat 1 is breeding sites *1n The population census is taken immediately prior to the breeding season. Each adult produces β juveniles that are alive at the end of the breeding season. There is no adult mortality during breeding. Adults survive the winter with probability PA and juveniles survive with probability PB nnβ+WINTEREnd of WINTERANNUAL CENSUSSUMMER(reproductive Season)nPnPnJAβ+='End of SUMMERn Population growth can be described by the following model 11'1nnPnPnJAλβ=+=Source Habitat Remember that the population is limited by the number of breeding sites in habitat 1 and will grow until it reaches its limit . Once all breeding sites are occupied then the habitat specific population growth can be described by: *1n *11'1nnλ= However, only will stay in habitat 1, the remainder of individuals with emigrate. *1n Since 11>λ (by definition) then ()11−λ is the per capita surplus being generated by the source and ()*111 n−λ individuals will emigrate Sink Habitat By definition 12<λ and the population would eventually decline to zero, unless supported by the sink habitat; therefore, the population dynamics in the sink can be described by: 2122'2inn +=λ where is immigration from the source habitat. 21i Let = the sink population size at equilibrium. The population will reach equilibrium when immigration from the source equals the decline in the sink. *2n ()21*21 in =−λ Since immigrants are coming from the surplus produced by