AboutZoos, Since 2008


Species per­sis­tence or extinc­tion: Through a math­e­mat­i­cal lens

pub­lished 14 Novem­ber 2012 | mod­i­fied 04 Decem­ber 2012
African wild dog PeaugresSci­en­tists have esti­mated that there are 1.7 mil­lion species of ani­mals, plants and algae on earth, and new species con­tinue to be dis­cov­ered. Unfor­tu­nately, as new species are found, many are also dis­ap­pear­ing, con­tribut­ing to a net decrease in bio­di­ver­sity. The more diver­sity there is in a pop­u­la­tion, the longer the ecosys­tem can sus­tain itself. Hence, bio­di­ver­sity is key to ecosys­tem resilience.

The causes for dis­ap­pear­ing species are many, such as dis­ease, destruc­tion of habi­tats, pol­lu­tion, chem­i­cal and pes­ti­cide use, increased UV-​B radi­a­tion, and even the pres­ence of new species. The phe­nom­e­non known as “Allee effect,” by which a population’s growth declines at low den­si­ties, is another key rea­son for per­ish­ing pop­u­la­tions, and is an over­rid­ing fea­ture of a paper pub­lished last month in the SIAM Jour­nal on Applied Math­e­mat­ics.

Authors Avner Fried­man and Abdul-​Aziz Yakubu use math­e­mat­i­cal mod­el­ing to analyse the impact of dis­ease, ani­mal migra­tions and Allee effects in main­tain­ing bio­di­ver­sity. Some Allee effect causes in smaller and less dense pop­u­la­tions are chal­lenges faced in find­ing mat­ing part­ners, genetic inbreed­ing, and coop­er­a­tive behav­iours such as group feed­ing and defense. The Allee thresh­old in such a pop­u­la­tion is the pop­u­la­tion below which it is likely to go extinct, and above which per­sis­tence is pos­si­ble. Declin­ing pop­u­la­tions that are known to exhibit Allee effects cur­rently include the African wild dog and the Florida pan­ther.

Abdul-​Aziz Yakubu explains how dis­ease can alter the behav­iour of pop­u­la­tions that exhibit Allee effects. In infec­tious dis­ease stud­ies, the repro­duc­tion num­ber or Ro is defined as the expected num­ber of sec­ondary infec­tions aris­ing from an ini­tial infected indi­vid­ual dur­ing the latter’s infec­tious period. For reg­u­lar pop­u­la­tions, the dis­ease dis­ap­pears in the pop­u­la­tion if (and only if) the Ro is less than 1. “In the present paper, we deal with a pop­u­la­tion whose sur­vival is pre­car­i­ous even when Ro is less than 1,” says Yakubu. “That is, inde­pen­dent of Ro, if the pop­u­la­tion size decreases below a cer­tain level (the Allee index), then the indi­vid­u­als die faster than they repro­duce.”

A pre­vi­ous study by the authors showed that even a healthy sta­ble pop­u­la­tion that is sub­ject to Allee effects would suc­cumb to a small num­ber of infected indi­vid­u­als within a sin­gle loca­tion or “patch,” caus­ing the entire pop­u­la­tion to become extinct, since small per­tur­ba­tions can reduce pop­u­la­tion size or den­sity to a level below or close to the Allee thresh­old.

Trans­mis­sion of infec­tious dis­eases through a pop­u­la­tion is affected by local pop­u­la­tion dynam­ics as well as migra­tion. Thus, when try­ing to under­stand the resilience of the ecosys­tem, the global sur­vival of the species needs to be taken into account, that is, how does move­ment of ani­mals between dif­fer­ent loca­tions affect sur­vival when a dis­ease affects one or more loca­tions? Var­i­ous infec­tious dis­ease out­breaks, such as the West Nile virus, Phocine and dis­tem­per viruses have been seen to spread rapidly due to migra­tions.

In this study, the authors extend their pre­vi­ous research by using a multi-​patch model to analyse Allee effects within the con­text of migra­tion between patches. “We inves­ti­gate the com­bined effect of a fatal dis­ease, Allee effect and migra­tion on dif­fer­ent groups of the same species,” Yakubu says. In their con­clu­sions, the host pop­u­la­tion is seen to become extinct when­ever the ini­tial host pop­u­la­tion den­sity on each patch is lower than the small­est Allee thresh­old. When the ini­tial host pop­u­la­tion has a high Allee thresh­old, the pop­u­la­tion per­sists on each patch if the dis­ease trans­mis­sion rates are small and the growth rate is large. Even in the case of high Allee thresh­olds, the host pop­u­la­tion goes extinct if the dis­ease trans­mis­sion rate is high, and growth rate and dis­ease thresh­old are small. The pres­ence of a strong Allee effect adds the pos­si­bil­ity of pop­u­la­tion extinc­tion even as the dis­ease dis­ap­pears.

The research can be applied to var­i­ous kinds of pop­u­la­tions for con­ser­va­tion stud­ies.
Our mod­els and results are very gen­eral and may be applied to sev­eral declin­ing pop­u­la­tions. For exam­ple, the African wild dog, an endan­gered species, is vul­ner­a­ble to fatal dis­eases like rabies, dis­tem­per and anthrax. Our mod­els can be used to inves­ti­gate how the Allee thresh­old of one sub­pop­u­la­tion of an African wild dog pack at a geo­graph­i­cal loca­tion is influ­enced by the col­lec­tive migra­tions of sev­eral wild dog pop­u­la­tions from dif­fer­ent packs with dif­fer­ent Allee thresholds
(Pro­fes­sor Abdul-​Aziz Yakubu, Depart­ment of Math­e­mat­ics, Howard Uni­ver­sity, Wash­ing­ton, DC)

The authors’ math­e­mat­i­cal mod­els and rig­or­ous analy­sis can be extended with the help of field data. “Future work will need to get spe­cific field data in order to refine the model and use it to design con­ser­va­tion strate­gies for preser­va­tion of these some­what endan­gered and declin­ing pop­u­la­tions,” says Yakubu

The above news item is reprinted from mate­ri­als avail­able at Soci­ety for Indus­trial and Applied Math­e­mat­ics. Orig­i­nal text may be edited for con­tent and length.
(Source: SIAM News, 12.11.2012)
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