Allee effects in randomly varying environments

Proceedings of the ²⁰¹³

Intemational Confetence on Computational ^and Mathematical Methods in Science and Engineering

Cabo de Gata, Almeria, ^SPain

June24-27,2013

Co and Mathematical Methods ^{ence and} Engineering

Editors

I. P. Hamilton & J. Vigo-Aguiar

Associate Editors

H. Adeli, P. Alonso, M.T. De Bustos,

M. Demiralp, J.A. Ferreira, A. _Q. M. Khaliq,

J.A. L6pez-Ramos, P. Oliveira, J.C. Reboredo,

M. Van Daele, E. Venturino, J. Whiteman, B. Wade

rsBN

978-84-6 16-27

23-3

@Copyright 2013 CMMSE Printed on acid-free ^paPer

@CMMSE Preface- Page ii

Prottr:rLinqs of th,e ^1.9t.h Int.entation.al Conferutcc ort Conytutati,onal a,n,d, Mathemntical Mcthods in Sr:itrtce ond, Enqineering. CIUIMSE 201,1

24 27,1lurc, ^201:J.

Allee effects models in randomly varying environments

Carlos A. Braumannl and Clara

Carlos2

I

_Cetl|lo d,e ltn)estiaa1ao cm l[rttttn.dtittr e Altlir:ar;ot:s. u-nit'tt sitla.lt de Etttttt 2 Esu ,a Sqx'.r'ilr dc'lbcnol,ogio, do Bottr:i,ro. lnst'ituto Poli.t.&:nico de Setihol

clnails: braumam@uevora.

pt. clara.

carlos@estbarreiro ^.

iPs.pt

Abstract

llascrl on a deterll nistic nlodel ol lnpulalion ^grorvtb wilh Alltt cllccts. we pr()lx)se 1qcncrn.l stochasl,ic ntrxlel that incorporates cnvironnrcntal taudotrt Ilrrctuations ^{irr the} grorvth |roccss. ^\Vc study the tnodel propt'rlit:s. exislencc lld ttttiqueness of sohlli()Ir arld the stationary bchavior. We also obtain exptessiorrs lbr f'h' firsl' passagc tilrxrs, in particular. thc rncau arul slandard dcviation ol exli clion litlrcs ior lhe popttlatiolt.

Keu utords: AIIee elJects. Inplrldli.ott Qronlh ntndortt cnvirornnt:nts' erti'rt(tlon Itlttt's

1 Introduction

\\.a.r rlcr. clvdc Allcc (1885/1955) was an Arneri< an zoolrgist arrrl er:ologist q'ho ta[ght aninlal 0cology. Tlur Allec ^t:ftects wcrc fiLst desr:ribetl by Alk'0 and (ollcagues irt 19.19, a.s wc can sce

in

[2]. There ^are two kinds of Allec cffccts. strong Allcc effer:ts arrd l^('ak

Alk!

^{effr'< ts.}

lvhcrr a poprrlation ^has a ,'critical size or densitv" lrclow which thc poprllation ^dccrcases on aveLage atrcl above whir:h

it

irx:reascs on avcrage.

it

is called a strong Allcc oflect. orr the othcr hand. ^{q,'hcn a} popl atiorr does IIot cxhibit ^a "critit:al ^{sizc or} dcnsit-v"

l)tlt

at low cleDsities tlrc population growth rate increa.ses

with

irrcreasirrg dctrsity. wc sav tllele is ^a rvcak Allee cffect. Allee effects sho*'rrp

in

^rrranv u'ildlifc poprrlaliors, palticttlatlY wht'rl low populatiorr sizc hindcrs the cfficacy of collcc:tive (lefensive bchavior

fiorl

pr'odatols or l'esrrlts iD irrrlividuals being far apaJt, which nl?rkes

it

dilficlrlt to find mating ^{palt ueIS.}

\\'c

introduce a quite ^general deterministic model of

poplllatiut

growth $'itlr strong Allcc t,ffect, particrrlzrr ca.scs of which ^(:arr b(' ^seen

in

_[6],

lil].

[10]

or [1]. ^'Ihe

stocha.sti<r nro<lel

uc

proposc iri ba.scd on this deter.rnirristic

[xxlcl with

an arlded

tclm

tO ^account

for

thc efiect of environrncntal flrrctuations ou

tltc

^growth

rate. other

stochastic AIhe

ocMMSE tsBN 978 84-616-2723 ³