Abstract
Competition between species for resources is a fundamental ecological process,which can be modeled by the mathematical models in the chemostat culture or in the water column. The chemostat-type models for resource competition have been extensively analyzed. However, the study on the competition for resources in the water column has been relatively neglected as a result of some technical difficulties.We consider a resource competition model with two species in the water column. Firstly, the global existenceand $L^\infty$ boundedness of solutions to the model are established by inequality estimates. Secondly, the uniqueness ofpositive steady state solutions and some dynamical behavior of the single population model are attained by degree theory and uniformpersistence theory. Finally, the structure of the coexistence solutions of the two-species system is investigated bythe global bifurcation theory.
Highlights
Competition between species for resources is a fundamental ecological process[7, 20]
The study on resource competition in the water column has been relatively neglected as a result of some technical difficulties
By virtue of the complex boundary conditions, it is hard to establish the global existence of the solutions and a priori estimates of the positive steady state solutions
Summary
Competition between species for resources is a fundamental ecological process[7, 20]. If S(0) = 0, it follows from the existence and uniqueness of the solution to the ordinary differential equation that S ≡ 0, which is a contradiction to the boundary condition Sx(1) = β(1 − S(1)). For any δ0 > 0, suppose there exists a sequence (d(1i), d2(i)) ∈ [δ0, f1(1) − δ0] × [δ0, f2(1) − δ0] and positive solution (Si, ui, vi) to (7)-(8) with d1 = d1(i), d2 = d(2i) such that ui ∞ + vi ∞ → ∞ as i → ∞.
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