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A large number of herbivorous mammals and reptiles in many terrestrial ecosystems across the globe are presently in the receiving end of extinction. Over-exploitation by its immediate predator and anthropogenic actions is one of the main reasons. Reintroduction of apex predator or top predator at some instances has proven to be a successful strategy in restoring ecological balance. In this paper, we conceptualize the role of top predator in enriching the density of vulnerable species of lower trophic level, with the help of mathematical modeling. First, the dynamical behavior of two species system (prey and mesopredator) is studied, where growth of prey is subject to strong Allee effect. Also, the cost of predation induced fear is incorporated in the growth term. Parametric regions, for which the species perceive extinction risk are analyzed and depicted numerically. We consider that whenever density of the vulnerable species reach a certain threshold, minimum viable population, top predator is introduced in the habitat. Our obtained results show that a species population can be restored from the verge of extinction to a stable state with much higher population density with the introduction of top predator and even it stabilizes an oscillatory system. 

In this research, we study the impacts of the traceable mobility in a two-patch environment when the population in each patch exhibits strong Allee effects. Traveling individuals are traced across patches by budgeting the average time spent in each patch while keeping their place of residency. Particularly, we focus on the impact that the effective population (residents and visitors) produces on regional dynamics.Our results show that low mobility across regions produces simple dynamics, where orbits converge to single or double extinction, or to a coexistence steady state. We derive mobility conditions under which an endangered population may benefit of the presence of a visitant one and avoid extinction -- the rescue effect. Nonetheless, increments in the visiting population would also lead the resident population to extinction -- the induced extinction effect. 

In this paper, we investigate the dynamical behavior for a hybrid non-autonomous predator–prey system with Holling Type II functional response, impulsive effects and generalist predator on time scales, where our proposed model commutes between a continuous-time dynamical system and discrete-time dynamical system. By using comparison theorems, we first study the permanence results of the proposed model. Also, we established the uniformly asymptotic stability for the almost periodic solution of the proposed model. Finally, in the last section, we provide some examples with numerical simulation. 

The variation of nutrient supply not only leads to the differences in the phytoplankton biomass and primary productivity but also induces the long-term phenotypic evolution of phytoplankton. It is widely accepted that marine phytoplankton follows Bergmann's Rule and becomes smaller with climate warming. Compared with the direct effect of increasing temperature, the indirect effect via nutrient supply is considered to be an important and dominant factor in the reduction of phytoplankton cell size. In this paper, a size-dependent nutrient-phytoplankton model is developed to explore the effects of nutrient supply on the evolutionary dynamics of functional traits associated with phytoplankton size. The ecological reproductive index is introduced to investigate the impacts of input nitrogen concentration and vertical mixing rate on the persistence of phytoplankton and the distribution of cell size. In addition, by applying the adaptive dynamics theory, we study the relationship between nutrient input and the evolutionary dynamics of phytoplankton. The results show that input nitrogen concentration and vertical mixing rate have significant effects on the cell size evolution of phytoplankton. Specifically, cell size tends to increase with the input nutrient concentration, as does the diversity of cell sizes. In addition, a single-peaked relationship between vertical mixing rate and cell size is observed. When the vertical mixing rate is too low or too high, only small individuals are dominant in the water column. When the vertical mixing rate is moderate, large individuals can coexist with small individuals, so the diversity of phytoplankton is elevated. We predict that reduced intensity of nutrient input due to climate warming will lead to a trend towards smaller cell size and will reduce the diversity of phytoplankton.