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Abstract The global dynamics of the two-species Lotka–Volterra competition patch model with asymmetric dispersal is classified under the assumptions that the competition is weak and the weighted digraph of the connection matrix is strongly connected and cycle-balanced. We show that in the long time, either the competition exclusion holds that one species becomes extinct, or the two species reach a coexistence equilibrium, and the outcome of the competition is determined by the strength of the inter-specific competition and the dispersal rates. Our main techniques in the proofs follow the theory of monotone dynamical systems and a graph-theoretic approach based on the tree-cycle identity. 

Restoration of native oyster ( Crassostrea virginica ) populations in Chesapeake Bay shows great promise after three decades of failed attempts. Population models used to inform oyster restoration had integrated reef habitat quality, demonstrating that reef height determines oyster population persistence and resilience. Larval recruitment drives population dynamics of marine species, yet its impact with reef height and sediment deposition upon reef restoration is unknown. To assess the influence of reef height, sediment deposition and larval supply, we adapted a single-stage population model to incorporate stage structure using a system of four differential equations modeling change in juvenile density (J), and changes in volume of adults (A), oyster shell reef (R), and sediment (S) on an oyster reef. The JARS model was parameterized with empirical data from field experiments. Larval supply included larvae from the natal population and from outside populations. The stage-structured model possessed multiple non-negative equilibria (i.e., alternative stable states). Different initial conditions (e.g., oyster shell reef height) resulted in different final states. The main novel findings were that the critical reef height for population persistence and resilience was jointly dependent on sediment input and larval supply. A critical minimum larval supply was necessary for a reef to persist, even when initial sediment deposition was zero. As larval supply increased, the initial reef height needed for reef persistence was lowered, and oyster reef resilience was enhanced. A restoration oyster reef with higher larval influx could recover from more severe disturbances than a reef with lower larval influx. To prevent local extinction and assure a positive population state, higher levels of larval supply were required at greater sediment concentrations to overcome the negative effects of sediment accumulation on the reef. In addition, reef persistence was negatively related to sediment deposited on a reef prior to larval settlement and recruitment, implying that restoration reefs should be constructed immediately before settlement and recruitment to minimize sediment accumulation on a reef before settlement. These findings are valuable in oyster reef restoration because they can guide reef construction relative to larval supply and sediment deposition on a reef to yield effective and cost-efficient restoration strategies.