Stano Pekár“Populační ekologie živočichů“  dN = Nr dt Spatial ecology - describes changes in spatial pattern over time  processes - colonisation / immigration and local extinction / emigration  local populations are subject to continuous colonisation and extinction  wildlife populations are fragmented Metapopulation - a population consisting of many local populations (sub-populations) connected by migrating individuals with discrete breeding opportunities (not patchy populations)  population density changes also in space  for migratory animals (salmon) seasonal movement is the dominant cause of population change  movement of individuals between patches can be density-dependent  distribution of individuals have three basic models:  most populations in nature are aggregated (clumped) Regular distribution  described by hypothetical discrete uniform distribution n .. is number of samples x .. is category of counts (0, 1, 2, 3, 4, ...)  all categories have similar probability  mean:  variance:  for regular distribution: n xP 1 )(  )1( 2 1  n )1( 12 1 22  n 2    described by hypothetical Poisson distribution  .. is expected value of individuals x .. is category of counts (0, 1, 2, 3, 4, ...)  probability of x individuals at a given area usually decreases with x  observed and expected frequencies are compared using 2 statistics  for random distribution: ! )( x e xP x     Random distribution 2    described by hypothetical negative binomial distribution  .. is expected value of individuals x .. is category of counts (0, 1, 2, 3, 4, ...) k .. degree of clumping, the smaller k (0) the greater degree of clumping  approximate value of k:  for aggregated: Coefficient of dispersion (CD) CD < 1 … uniform distribution CD = 1 … random distribution CD > 1 … aggregated distribution xk kkx xk k xP                   )!1(! )!1( 1)(     2 2 k Aggregated distribution x s CD 2  2   • species occupy new areas • expansion – increase in geographic range of native species • invasion - increase in geographic range of allien species Elton 1958 Spread of muskart in Europe ~ 5 km/year • individual makes blind random walk • random walk of a population undergoes diffusion in space • diffusion (Brownian motion) model in 2dimensional space: where D .. diffusion coefficient (distance2/time) - radial distance is related to time - slope of the line is rate of expansion               2 2 2 2 y N x N D t N time0 area N t=0 t=1 t=2 t=3           DtDt N tN 4 exp 4 ),( 2 0    Pure dispersal radius Dt4 t D 4 2   • Diffusion model - solved to 2dimensional Gaussian distribution - assuming all individuals are dispersers - range expanses linearly with time - no reproduction N0 .. initial density ρ .. radial distance from point of release (range) 0 N t=0 t=1 t=2 t=3 Dispersal + population growth radius rDc 2 0          Dt rt Dt N tN 4 exp 4 ),( 2 0    • Skellam‘s model - includes diffusion and exponential population growth r .. intrinsic rate of increase c - expansion rate [distance/time] - for species with R0 < 1.5 …. Skellam 1951 )ln(2 0R T c   Van den Bosch et al. 1990  types of spatially distributed populations: - Source-sink (mainland-island), panmictic (clumped), nonequilibrium (fragmented), metapopulation  the degree of isolation may vary depending on the distance among patches  unlike growth models that focus on population size, metapopulation models concern persistence of a population - ignore fate of a single subpopulation and focus on fraction of sub-population sites occupied Metapopulation assumptions: - Suitable habitat is in discrete patches - Population is reproductively active - Subpopulations have high risk of extinction - Subpopulations are not too isolated such that recolonization is likely - Dynamic are not synchronised across subpopulations - All patches are alike for extinction and colonization  Levins (1969) distinguished between dynamics of a single population and a set of local populations which interact via individuals moving among populations Hanski (1997) developed the theory - suggested coresatellite model  assumptions - sub-populations are identical in size, distance, resources, etc. - extinction and colonisation are independent of p - many patches are available - natality and mortality is ignored eppmp t p  )1( d d Levin‘s model p .. proportion of patches occupied m .. colonisation (immigration) rate - proportion of open sites colonised per unit time e .. extinction (emigration) rate - proportion of sites that become unoccupied per unit time Levin (1969) Time p 0.5 0.1  equilibrium is found for - sub-populations will persist (p* > 0) only if colonisation is larger than extinction (m > e) - Equilibrium state will always include empty patches if if e > 0 - K ..is fraction of patches - defined by balance between m and e  metapopulations are found in insects, Daphnia, frogs, birds rescue-effect model, incidence model m e m em p    1* K 0 d d  t p