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The distributions of species in their habitats are constantly changing. This phenomenon is thought to be determined by species’ environmental tolerance and biotic interactions for limited resources and space. Consequently, predicting the future distribution of species is a major challenge in ecology. To address this problem, we use mathematical model to study the combined effects of biotic interactions (e.g. competition) and environmental factors on multiple species community assembly in a heterogenous environment. To gain insights into the dynamics of this ecological system, we perform both analytical and numerical analyses of the range margins of the species. We observe that the range margins of the species can be influenced by biotic interactions combined with environmental factors. Depending on the strength of biotic interactions, our model exhibits coexistence of species and priority effects; mediated by weak and intense biotic interactions respectively. We also show the existence of bifurcation points (
*i.e.* the threshold values of competition coefficient) which lead to the presence—absence of different species. Thus, we suggest that adequate knowledge of biotic interactions and changes in the environments is important for effective maintenance of biodiversity and conservation management.

Ecologists have long been interested in answering the fundamental question of what determines the geographical distributions of species [

Another crucial force that can affect species distributions and range margins, is environmental factors in the form of climate change. This is because species have distinct responses to environmental changes and biotic interactions [

In addition, the order in which species become established may alter community assembly through priority effects (i.e. the effects one species have on community developments due to early arrival to site). In this case, the outcomes of species interactions depend on the initial abundances of the species; mediated by intense biotic interactions, such that species coexistence is impossible [

However, theoretical studies that analyze the combined effects of biotic and abiotic factors on multispecies dynamics with priority effects are rare. Thus, there is more to know about multispecies dynamics, when competition interactions occur in communities of more than two or three species in a geographically changing environment. According to May and Leonard [

Thus, we extend the deterministic model of Godsoe et al. [

We consider a multispecies deterministic model which is an extension of Lotka-Voltera competition model. The model is a system of ordinary differential equations for the densities N_{i}(t, x) of n species extended along one dimensional environmental gradient x, as in Equation (1) where 0 ≤ x ≤ 1.

d N i d t = r i N i k i ( x ) ( k i ( x ) − ∑ j = 1 n α i j N j ) ; ( i = 1 , 2 , ⋯ , n ) (1)

Here r_{i} is the intrinsic growth rate, k_{i}(x) is the carrying capacity, α_{ij} is the competitive strength of species j on species i, α_{ii} is the intraspecific competition coefficients (i.e. a measure of the strength of competition within the same species) and N_{i} is the densities of species i at time t. To make the model simple, we set intraspecific competition coefficients α_{ii} to 1. Thus, interspecific competition coefficients, α_{ij} represent the ratio of interspecific to intraspecific competition coefficients. In this study, we consider competition of four species (i.e. n = 4) and assume that competition strengths of the species are symmetrical (i.e. α_{ji} = α_{ij} = α). Then, Equation (1) becomes a system of four ordinary differential equations, one for each species with competition coefficient α.

The environmental suitability is modelled into the carrying capacity, k_{i}(x) of the species as a spatial dependence on the locations x (i.e. the carrying capacity, k_{i}(x) vary with the locations x). In this case, the environmental location is represented by x; which is used as a proxy to represent environmental factors like temperature, moisture, salinity and any other environmental factors that may affect the species. Therefore, the effects of biotic interactions on the species may depend on how the species respond to environmental factors. To show these effects, the carrying capacity is modelled such that it varies linearly with x.

k i ( x ) = m i x + b i (2)

In Equation (2), b_{i} is the species i carrying capacity at x = 0; m_{i} denotes the changes in the suitability of the environments with respect to abiotic component x (i.e. the gradient of the carrying capacity) and k_{i}(x) represents the carrying capacity of the species i at each location x. In this case, the maximum density that species i can attain is at x = 1 if m_{i} is positive.

To understand the dynamics of the system, ( d N i ) / d t in Equation (1) is set to zeros and we solved numerically for the steady states using MAPLE package. The stability analysis of the steady states is also carried out using MAPLE package. Thus, at a location x, the steady state who’s all the real parts of the eigenvalues are negative is stable. Based on the steady states and the numerical simulations results on the range margins, we used the techniques of invasion analysis to derive analytical results on the species’ range margins of the Equation (1). We also show from the invasion points of the species, the threshold values of the competition coefficient which can lead to presence-absence of different species across the locations x. The numerical simulation results on the range margins of the species are obtained by employing MATLAB ode15 solver for t = 1000 to solve Equation (1) until steady states are achieved. We also generate numerical simulation for the summary plot, using MATLAB ode15 solver in order to show different species present and their range margins across the geographical locations x as a model parameter, α changes. MAPLE package is also used to verify that the simulation results are stable. To further cross check the simulation results, numerical simulation package XPPAUT is used. Thus, the steady states of Equation (1) is computed with the aid of cvode solver for t = 1000. We then continued the steady states in AUTO; where the stable and unstable steady states as well as the bifurcation points are tracked as α changes. A parameter value, 10^{−1}/10^{−6} is used as the maximum/minimum allowable step size. Other parameter values that are used in the computation of the results are given in

Symbol | Items description | Parameter value |
---|---|---|

r_{i} | Intrinsic growth rate of species i | 1 |

m_{1} | Gradient of k_{1} | 1 |

m_{2} | Gradient of k_{2} | 0.8 |

m_{3} | Gradient of k_{3} | 0 |

m_{4 } b_{1 } b_{2 } b_{3 } b_{4} | Gradient of k_{4 } Carrying capacity of species 1 at x = 0 Carrying capacity of species 2 at x = 0 Carrying capacity of species 3 at x = 0 Carrying capacity of species 4 at x = 0 | 0 0 0 0.5 0.4 |

The analytical results of Equation (1) are presented in this section to show the range margins and threshold values of competition coefficient ( α T i ) of the species across locations x. The analytical results of the Equation (1) are derived using the method of invasion analysis to obtain the invasion points, x_{i} (i.e. the positions across the locations x where species i can invade when rare). The method uses the criterion that a species that can invade at a location x, must be rare at that point and its growth rate must be greater than zero (i.e. ( d N i ) / d t > 0 ) [_{1}(x) = m_{1}x, k_{2}(x) = m_{2}x, k_{3}(x) = b_{3} and k_{4}(x) = b_{4} respectively.

Based on _{1}) is taking to be zero. In a similar way, the density of species 2 (i.e. N_{2}) is also considered to be zero; since at the invasion point of species 1 only species 3 and 4 are present. Thus, species 1 can invade if k 1 ( x ) > α N 3 * + α N 4 * , where N 3 = N 3 * and N 4 = N 4 * . Therefore, the point x where:

k 1 ( x ) = α N 3 * + α N 4 * , (3)

satisfies species 1 invasion point (x_{1}). But k_{1}(x) = m_{1}x and for stable steady state, ( 0 , 0 , N 3 * , N 4 * ) = ( 0 , 0 , ( α k 4 − k 3 ) / ( α 2 − 1 ) , ( α k 3 − k 4 ) / ( α 2 − 1 ) ) , where k_{3} = b_{3} and k_{4} = b_{4}, which on substitution into Equation (3), species 1 invasion point is given as:

x 1 = α 2 ( b 3 + b 4 ) − α ( b 3 + b 4 ) m 1 ( α 2 − 1 ) (4)

Equation (4) implies that there exists an asymptote (i.e. x_{1}→∞) in the range margin of species 1 whenever α = 1 (i.e. the range margin of species 1 tends to infinity at α = 1). Thus, based on the Equation (4), the threshold value of competition coefficient of species 1 in Equation (6) can be established through Equation (5).

m 1 x 1 ( α 2 − 1 ) = ( α 2 − α ) ( b 3 + b 4 ) (5)

Thus,

α T 1 = m 1 x ( b 3 + b 4 ) − m 1 x (6)

Moving along the environmental gradient x to the right of x_{1} (see _{2}) of species 2 can be derived like species 1. In this case, ( d N 2 ) / d t in Equation (1) must be greater than zero for species 2 to invade. Also, at the invasion point of species 2, its density (i.e. N_{2}) is also, considered equivalently zero. Thus, species 2 can invade if k 2 ( x ) > α N 1 * + α N 3 * + α N 4 * , where = N 1 = N 1 * , N 3 = N 3 * and N 4 = N 4 * . Then, species 2 invasion point satisfies that:

k 2 ( x ) = α N 1 * + α N 3 * + α N 4 * , (7)

where k_{2}(x) = m_{2}x, k_{3} = b_{3}, k_{4} = b_{4} and the steady state is given as:

( N 1 * , 0 , N 3 * , N 4 * ) = ( ( α 2 ( k 3 + k 4 − k 1 ) − α ( k 3 + k 4 ) − k 1 ) / ( 2 α 3 − 3 α 2 + 1 ) , 0 , ( α 2 ( k 1 + k 4 − k 3 ) − α ( k 1 + k 4 ) − k 3 ) / ( 2 α 3 − 3 α 2 + 1 ) , α 2 ( k 1 + k 3 − k 4 ) − α ( k 1 + k 3 − k 4 ) / ( 2 α 3 − 3 α 2 + 1 ) ) .

Thus, substituting into Equation (7), gives the invasion point of species 2 as in Equation (8).

x 2 = α 3 ( b 3 + b 4 ) + ( 1 − 2 α ) α ( b 3 + b 4 ) m 2 ( 2 α 3 − 3 α 2 + 1 ) − α m 1 ( α 2 − 2 α + 1 ) (8)

Like species 1, the threshold value of competition coefficient of species 2 becomes:

α T 2 = m 2 x ( b 3 + b 4 ) + m 1 x − 2 m 2 x (9)

Based on Equation (8), the first scenarios by which species 2 range margin is likely to increase depends on strong interspecific competition from species 3 and 4 at the boundary of species 2 fundamental niche. For instance, larger values of α(b_{3} + b_{4}) can shift species 2 invasion point (x_{2}) and then, increase the range of x for which species 2 can be absent. Therefore, a modest change in the model parameters will also cause a modest change in the range margin of species 2 [_{1}). As the denominator tends to zero, there exists an asymptote (i.e. x_{2 }→∞). In this scenario, small changes in the model parameters will result to high changes in the invasion point (x_{2}) of species 2 [

In a similar way, species 4 invasion point can be computed in the presence of species 1, 2 and 3. Thus, invasion point of species 4 is given as:

x 4 = b 4 ( 2 α 3 − 3 α 2 + 1 ) − α b 3 ( α 2 − 2 α + 1 ) ( α 2 − 2 α + 1 ) α ( m 1 + m 2 ) (10)

The threshold value of competition coefficient of species 4 is given as in Equation (11).

α T 4 = b 4 ( m 1 + m 2 ) x − 2 b 4 + b 3 (11)

At the location x = 0.5 for instance, the threshold value of competition coefficient of species 4is equal to the threshold value of competition coefficient of species 2. This implies simultaneous extinction of species 4 and 2 at the same competition strength and location.

Also, the invasion point of species 3 is computed to give:

x 3 = b 3 ( α 2 − 1 ) ( α − 1 ) α ( m 1 + m 2 ) (12)

Like species 1, the range margin of species 3 tends to infinity if α = 1 in Equation (12). This implies that the range margins of species 1 and 3 both tend to infinity at α = 1; since x_{3 }= x_{1} at the value of α = 1. Based on the Equation (12), the threshold value of competition coefficient of species 3 is given as in Equation (13).

α T 3 = b 3 ( m 1 + m 2 ) x − b 3 (13)

The threshold value of competition coefficient of species 3 is also, equal to the threshold value of competition coefficient of species 1 at the location x = 0.5 and α = 1.25. See Equation (6) and parameter values given in

Numerical results are presented in this section to illustrate the influence of biotic interactions and environmental gradients on the range margins of species across locations x. Both the numerical and analytical results agreed with each other (compare

To show the impacts of competitive strengths on multispecies community assembly, the numerical results are obtained separately for different competitive strengths of the species. For example, when competitive strengths of the species are relatively weak (see

presence-absence of the species. A detection threshold value of 0.5% of the maximum observed density of the species is employed for the numerical results presented in this section. This means that a species is considered absent if its density is below this expected value [

In _{1} = 0.3375 and x_{2} = 0.4655 respectively. Similarly, species 3 and 4 are shifted by species 1 and 2 on the right side of the locations x;such that only species 1 and 2 coexisted in the domain. Hence, species 4 and 3 range margins are shifted to a point x_{4} = 0.5370 and x_{3} = 0.7407 respectively. The multispecies coexistence at the central region, illustrates the most favorable locations in the environments. The range margins shown in

To illustrate the impact of the environmental gradients, we computed _{2}(x) = 0.8x to k_{2}(x) = 2x – 0.8. Other parameter values remain as in _{i}) of the species changed to x_{1}, x_{4}, x_{2} and x_{3} in contrast to community assembly observed in

In this section, we investigate the effects of intense biotic interactions (i.e. α > 1) on competition outcomes among the species. In this case, we observed that coexistence of species in one location is impossible due to aggressive interactions. As a result, this situation leads to occurrence of priority effects; and the dynamical model’s behavior is such that species’ initial abundances determine the outcomes of the competitions. Ecologically, priority effects can be referred to as alternative stable states; a situation where the presence-absence of the species depends on the order of arrival to site [_{2}(x) = 0.75k_{2}(x); and in the third row (_{4}(x) = 0.75k_{4}(x). This implies that in _{1}(x) = 0.1k_{1}(x), N_{2}(x) = 0.9k_{2}(x), N_{3}(x) = 0.1k_{3}(x), N_{4}(x) = 0.9k_{4}(x); in _{1}(x) = 0.1k_{1}(x), N_{2}(x) = 0.75k_{2}(x), N_{3}(x) = 0.1k_{3}(x), N_{4}(x) = 0.9k_{4}(x) and in _{1}(x) = 0.1k_{1}(x); N_{2}(x) = 0.9k_{2}(x); N_{3}(x) = 0.1k_{3}(x); N_{4}(x) = 0.75k_{4}(x). In each case, the spatial domain is partitioned into four regions, depending on the initial abundances of the species (see

Throughout, species 2 and 4 are more abundant and so, have higher potential to exclude species 1 and 3 from some locations and occupied a larger domain. However, species 3 and 4 dominated the left side of the domain (see Figures 2(a)-(f)) and excluded species 1 and 2 in the region. Species 4 with higher initial abundance compared to species 3 dominated the larger part of the left region and shifted species 3 to a narrower central region. Similarly, species 1 and 2 dominated the right side of the domain (see Figures 2(a)-(f)) and exclude species 3 and 4. Species 2 with initial abundance advantage, also dominated the larger part of the right domain and then shifted species 1 towards a smaller right center. We observed that the decrease in species 2 initial abundance (see second row) tipped a balance more for species 1 and increase its region of dominance. Similarly, the reduction of initial abundance of species 4 (see third row) tipped a balance more for only species 3. These observations show that in a multispecies community dynamic, initial abundance can determine the range margins of the species; but ecologically similar species may likely have more impacts on one another than the reverse.

When competition coefficient, α < 1, multispecies coexistence (red colored region) is observed at the center of the location x. Due to changes in the environments and biotic interactions, the number of species coexistence decreases as one move away from the center. These observations are also, illustrated in

To further understand the different species presence-absence in the summary plot, we employed numerical continuation to compute _{t}_{1}) and saddle node bifurcation (α_{T}_{2}) points in _{t}_{1} (compare _{t}_{1} is also, consistent with species 2 and 4 threshold values of competition coefficients at the location x = 0.5 (see _{t}_{1} < α < α_{T}_{2}); with simultaneous exclusion of species 2 and 4. The vertical red line at α = 1 indicate an asymptote, where the range margins of species 1 and 3 tend to infinity (refer to Equations 4 and 12). Beyond α > α_{T}_{2}, are another stable steady state branch of species 1 and 3, each existing as a single-species to the maximum density of the carrying capacity. This is separated by unstable steady state (black line) of two species coexistence. Based on our analytical results (refer to Equations (6 and 13)) and the summary plot (

We investigate the dynamical outcomes of multispecies competition in an environmentally changing habitat. One of the significant results we observed is that when biotic interactions are relatively weak, species can coexist, with multispecies coexistence near the center. This form of community assembly has earlier been observed in an empirical study of small mammal species along elevational gradients [

It is also observed that when biotic interactions are severe (α_{j} > 1), this situation can lead to occurrence of priority effects. In this case, the initial abundances of the species determine the presence-absence or the range margins of the species. Ecologically, these qualities could be implemented in biocontrol management; for preserving some species of interest or to reduce the excesses of a species whose activities are undesirable in the habitat. As illustrated in

Also, environmental factors combined with biotic interactions are illustrated in our model results (compare _{i}(x) may engender different dynamical behaviors. However, the choice of k_{i}(x) which affects only species 3 and 4 carrying capacities, will qualitatively correspond to the dynamical behaviors illustrated in this study. For instance, changing the slope of the carrying capacities of species 3 and 4 from m_{3} = 0 to m_{3} < 0 (respectively from m_{4} = 0 to m_{4} < 0), will not change the invasion points of the species as much from the one illustrated in this study. This is because, we still have two pairs of ecologically similar species interacting with one another. In this case, a small variation in the steepness of species 3 and 4 carrying capacities, will lead to a modest change in the invasion point of the species [

Also, our numerical continuation results which illustrate both stable and unstable steady states and bifurcation points of the models, proffer detail explanation for the different species presence-absence observed in the numerical simulation results. Our model exhibits existence of threshold values of competition coefficient, α. The threshold values correspond to critical values (or color change) in the summary plot, where one combination of species presence exchanged its stability for another combination of species. The bifurcation points therefore, give rise to different dynamical behaviors of the model; such as coexistence, exclusion of species and priority effects.

In conclusion, we used deterministic model, which is a system of ordinary differential equations, to investigate the combined effects of competition interactions and environmental factors on multispecies community dynamics. The model is analyzed for the range margins of the species using analytical and numerical methods. Both analytical and numerical simulation results are found to agree with each other. The findings of this study have shown that biotic interactions and environmental factors can combine to determine the range margins of species. The results revealed different dynamical outcomes, which depending on the species’ competition strengths can be coexistence of species or priority effects. Thus, based on the findings, we suggest that adequate knowledge of biotic interactions and changes in the environments is essential for successful maintenance of biodiversity and conservation management. Also, ecological factors such as dispersal may change the outcomes of the competition dynamics presented in this study. We therefore, recommend dispersal inclusion in the deterministic model in this study; as an interesting extension which can lead to robust predictions of the range margins of species across a geographical region.

Thanks to the School of Mathematical Sciences and the Universitti Sains Malaysia for the support and encouragements.

The authors declare no conflicts of interest regarding the publication of this paper.

Omaiye, O.J. and Mohd, M.H. (2019) The Roles of Biotic Interactions and Environmental Factors on Multispecies Dynamics. Open Journal of Ecology, 9, 426-442. https://doi.org/10.4236/oje.2019.910028