Mathematical Model of the Dynamics of Fish, Waterbirds and Tourists in the Djoudj National Park, Senegal
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Mathematical Model of the Dynamics of Fish, Waterbirds and Tourists in the Djoudj National Park, Senegal Oumar Diop1 • Abdou Se`ne2
Received: 15 December 2015 / Accepted: 27 September 2016 Springer Science+Business Media Dordrecht 2016
Abstract In the present paper, we propose and analyze a harvested predator–prey model that incorporates the dynamics of tourists in the Djoudj National Park of Birds, Senegal. The model describes the impact of migration of waterbirds and seasonal fishing on the global coexistence of species in the site of the Djoudj. By the Mahwin continuation theorem of coincidence degree theory, we investigate the existence of a positive periodic solution. The global asymptotic stability is discussed by constructing a suitable Lyapunov functional. Some computational results are also addressed. Keywords Population dynamics Djoudj National Park Permanence Continuation theorem Global stability
1 Introduction The Djoudj National Park of Birds is a suitable wetland for migratory waterbirds. It receives more than 366 European species of migratory birds, particulary Palearctic and Afrotropical birds and is the fief of white pelican in reproduction (Tre´ca et al. 1993). It is one of the first fresh water sources they reach after crossing 200 km of the Sahara desert. From September to April, an estimated 3 millions migratory birds stay there. Regularly thousands of migratory birds like flamingo as well as white pelican, tree ducks, spur-winged goose, herons, egrets, spoonbill, African darter, cormorant nest there. This park is located in the delta of the Senegal river, near Senegal’s border with Mauritania. It covers some 16,000 ha of river channels, backwaters, streams, ponds and a large lake. The aquatic area covers an important class of fish harvested by & Oumar Diop [email protected] 1
LANI-UFR SAT, Universite´ Gaston Berger, B.P. 234, Saint-Louis, Senegal
2
CEA-MITIC, Universite´ Gaston Berger, B.P. 234, Saint-Louis, Senegal
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O. Diop, A. Se`ne
human and migratory waterbirds. The Djoudj is also a favored place for tourism thanks to its natural beauty and abundance of waterbirds. More than 14,000 tourists visit the park per year. For these reasons the park was listed as a Ramsar site in 1980 and accepted as a Unesco World Heritage site in 1981. Researchers both on mathematical and theoretical ecology proposed mathematical models to describe the relationship between tourism and environmental parameters (Lacitignola et al. 2007; Laurab and Leon 2006; Davis 1999). They investigated autonomous models of the growth of tourists in a given wetland and studied existence of equilibria and their stability. This is not realistic when environmental parameters are seasonal fluctuation. In non autonomous predator– prey models with seasonal variation of ecological parameters, researchers investigated the existence of periodic solutions (Moussaoui et al. 2015; Moussaoui and Bouguima 2016; Dai and Tang 1998; Ton and Hieu 2011; Kwang 2011; Freedman and Wu 1992). Most of them employ the Mahwin con
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