Classical Lotka-Volterra (LV) competition equation has shown that coexistence of competitive species is only possible when intraspecific competition is stronger than interspecific competition, i.e., the species inhibit their own growth more than the growth of the other species. Note that density effect is assumed to be linear in a classical LV equation. In contrast, in wild populations we can observed that mortality rate often increases when population density is very high, known as crowding effects. Under this perspective, the aggregation models of competitive species have been developed, adding the additional reduction in growth rates at high population densities. This study shows that the coexistence of a few species is promoted. However, an unsolved question is the coexistence of many competitive species often observed in natural communities. Here, we build an LV competition equation with a nonlinear crowding effect. Our results show that under a weak crowding effect, stable coexistence of many species becomes plausible, unlike the previous aggregation model. An analysis indicates that increased mortality rate under high density works as elevated intraspecific competition leading to the coexistence. This may be another mechanism for the coexistence of many competitive species leading high species diversity in nature.Competition is one of the fundamental ecological interactions between species 1 . We can observe that coexisting species are competing for the same resources 2 . A typical resource competition model which has been recognized widely is the Classical Lotka-Volterra competition 1,2 . The analyses of this equation show that coexistence of two or more species becomes only possible if intraspecific competition is stronger than interspecific competition 3 . Otherwise, dynamics leads to the exclusion of one species among n species, known as the competitive exclusion principle 1,4 . However, in natural communities many competing species have been coexisting in the same habitat over time, resulting in a high species diversity. Hence, we suspect that there should be some mechanisms for coexistence of competitive species, e.g., spatial structures 5,6 . These models, however, introduce an additional complexity into the mathematical models of classical LV systems. Compared with these complex models, coexistence of multiple species in natural communities seems to be far more ubiquitous. Therefore, a more universal explanation may be worth considering.Crowding effect is considered as one of the ubiquitous mechanisms in any biological populations [7][8][9][10][11][12][13][14][15][16] . A nonlinear density effect at high densities is called crowding effect, while that at low densities, Allee effect 7-17 . Unlike this nonlinear density effect, in the traditional mathematical models of population dynamics, density effect is

Predator-prey systems have been studied intensively for over a hundred years. These studies have demonstrated that the dynamics of Lotka-Volterra (LV) systems are not stable, that is, exhibiting either cyclic oscillation or divergent extinction of one species. Stochastic versions of the deterministic cyclic oscillations also exhibit divergent extinction. Thus, we have no solution for asymptotic stability in predator-prey systems, unlike most natural predator-prey interactions that sometimes exhibit stable and persistent coexistence. Here, we demonstrate that adding a small immigration into the prey or predator population can stabilize the LV system. Although LV systems have been studied intensively, there is no study on the non-linear modifications that we have tested. We also checked the effect of the inclusion of non-linear interaction term to the stability of the LV system. Our results show that small immigrations invoke stable convergence in the LV system with three types of functional responses. This means that natural predator-prey populations can be stabilized by a small number of sporadic immigrants.

In grassland studies, an intermediate level of grazing often results in the highest species diversity. Although a few hypotheses have been proposed to explain this unimodal response of species diversity to grazing intensity, no convincing explanation has been provided. Here, we build a lattice model of a grassland community comprising multiple species with various levels of grazing. We analyze the relationship between grazing and plant diversity in grasslands under variable intensities of grazing pressure. The highest species diversity is observed at an intermediate grazing intensity. Grazers suppress domination by the most superior species in birth rate, resulting in the coexistence of inferior species. This unimodal grazing effect disappears with the introduction of a small amount of nongrazing natural mortality. Unimodal patterns of species diversity may be limited to the case where grazers are the principal source of natural mortality.