Competition promotes the persistence of populations in ecosystems

Competition is one of the most common form in ecological systems, which plays important roles in population dynamics. However, the influences of competition on persistence of populations remain unclear when space effect is included. In this paper, we investigated a predator-prey model with competition and spatial diffusion. Based on pattern formations and time series of populations, we found that competitions induce the persistence of populations, which denies competitive exclusion principle. Moreover, we testify the robustness of these effects. Our results also suggest that space may lead to the emergence of new phenomenon in ecosystems.


Results
In this part, we show our results based on a predator-prey model in reaction diffusion form (see Method section). In the first step, we display the spatial patterns in two cases and reveal the difference in these two cases. In the following step, we demonstrated competition would induce the persistence of populations. Finally, we revealed that the finding that persistence induced by competition is widely applicable in ecosystems.
Spatial patterns of populations. Pattern structures may provide some important information on the state of the populations including persistence or extinction. Spatial patterns in different types imply different biological meanings. Our simulations employ the zero-flux boundary conditions with a system size of 200 × 200 space units. Time step is Δ t = 0.0001 and space step is Δ h = 1. And the initial density distributions are random spatial distributions of the predator and prey populations. During the simulations, different types of dynamics are obtained and the distributions of predator and prey are always of the similar type. As a result, we can restrict our analysis of pattern formation to one distribution (in this paper, we show the spatial pattern of predator populations).
In order to declare our results, we show the spatial patterns of populations in two cases: (i) systems (7)-(8) without competition; (ii) systems (7)-(8) with competition. We perform a series of numerical simulations of the spatially extended systems (7)-(8) in two-dimensional spaces, and the dynamical behavior are represented by figures. In the first step, we show the spatial pattern of predator populations in systems (7)-(8) without competition (see Fig. 1) and the parameter sets are a = 1.3, b = 0.2, l = 1.1, m = 0.09 and e = 0. From Fig. 1, we can see that regular spiral waves emerge after the perturbation of the stationary solutions of the spatially homogeneous systems (7)- (8). After a long time, the spiral waves do not break up and stay stable in the two-dimensional space.
In the following step, we show the evolution of the spatial pattern of predator populations with competition at t = 100, 200, 300 and 1000 in Fig. 2 with e = 0.5. After the random conditions, spotted spatial patterns with different radius emerge. After some iterations, regular spots with the same radius prevail over the whole domain and the dynamics of the system does not undergo any further changes. Compared Fig. 1 with Fig. 2, one can conclude that competition in predator populations induces the pattern transition from no-stationary patterns (spiral waves) to stationary patterns (spotted patterns).

Competitions induce the persistence of populations.
To see the effects of competition on the persistence of populations, we use time series analysis to show our results. In Fig. 3, we present the average density of predator population when there is no competition. For a long time, the density will exhibit a periodic behavior. However, the minimum value is too small and the predator population may suffer a high risk of extinction. In the real world, populations live in the natural environment and thus they are strongly influenced by abiotic factors such as weather and climatic conditions 30,31 . Under these circumstances, populations will be induced to be extinct   where n c means the number of points when predator density is smaller than the critical density, and n total is the total number points we employed. In Fig. 4, we found that the population may extinct if critical value is small. That is to say, if the population has spiral wave, then it is apt to die out. When there is competition in predator population, the density of predator population will keep stable for a long time which is shown in Fig. 5. From this figure, we can find that the time series of the spatially averaged values of the population density shows that in the first intervals these values change fast as time increases. One can see that at t ≈ 350, the density of predator population reaches a constant value and the value increases much slower. As time further increases, the predator density has almost completely evolved and reaches its stable state. Based on time series analysis, we can draw a conclusion that competition may induce the persistence of populations.

Robustness of persistence induced by competition.
In the above part, we focused our attentions on the deterministic predator-prey system. We want to check that whether persistence of populations induced by competition can be found in stochastic environments. By numerical results on systems (18)- (19) (see Method section), we find that wave pattern can also be obtained if competition is not considered. Moreover, if critical value is small, then the populations may extinct with high rate. However, when competition is taken into account, then the population will go towards to stable state. Additionally, we further checked that in predator-prey systems with Holling type (such as Holling-II and III), competition can also promote the persistence of populations.

Discussion
In this paper, we investigated a predator-prey model with competition and spatial diffusion. It was found that competition may induce the persistence of populations. In the previous work, the studies on two competitive species showed that the species can not survive at the same time or in the same niche [12][13][14][15] . Our results suggest that competitive exclusion principle does not hold when space is included, which well extend the findings in the fields of population dynamics. At the same time, our results imply that space may cause the emergence of new phenomenon and thus space should not be ignored in the investigations of ecosystems.
It should be noted that our results are obtained based on a predator-prey model in the reaction-diffusion form. In this sense, we need to check that whether our results still hold if predator-prey models in other forms, such as cellular automata 32 . What is more, we also need to use the real data to verify our conclusions which will be well discussed in the future study.
For spatial diseases, they may also exhibit pattern transition from no-stationary patterns to stationary patterns, which suggests that the disease may persist as an endemic state 6,16 . In this case, we need to take measures to control the diseases. That is to say, the results obtained in this paper are widely applicable from ecosystems to epidemiology.

Method
A predator-prey model. We give three principal rules on our model: (1) In the absence of predator populations, prey populations grow logistically with intrinsic rate r and carrying capacity K; (2) Functional response depends on the densities of both prey (U) and predators (V) populations: (3) Competitions only occurs in predator populations. Based on the above rules, we obtain a predator-prey system with a ratio-dependent functional response and competitions which is as follows [33][34][35] : where U, V are prey and predator density, respectively. All parameters are positive constants, R stands for maximal growth rate of the prey, K carrying capacity, A capture rate, B conversion efficiency, D predator death rate, H handling time, and E competition rate. Δ = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 is the usual Laplacian operator in two-dimensional space and D 1 , D 2 are prey and predator diffusion coefficients. For sake of calculations, taking we arrive at the following equations containing dimensionless quantities (V is replaced by p): Emergence of spatial patterns. Denote E* = (n* , p* ) as the positive equilibria of systems (7)- (8). By linear analysis, we have the linear equations of systems (7)-(8): a n a p d n, and = + → ⁎ p p p r t ( , ), (12) into the systems (7)-(8) and assume < ⁎ n n and < ⁎ p p . The initial conditions are assumed as: , where the functions → l r ( ) and → z r ( ) decay rapidly for → → ± ∞ r . Following the standard approach, we perform a Laplace transformation of the linearized equations over the two independent variables → r and t. For where L(q) and Z(q) are the transforms of → l r ( ) and → z r ( ). As a result, we obtain the denominator: The condition for a spatial mode q (in one-or two-dimensional space) to be unstable and thus grow into spatial patterns is that Re(s) > 0.