A probabilistic approach to dispersal in spatially explicit meta-populations
Figure 2
Patch characteristics for NPC with (N=20). (a) individual patch connectivity versus normalized patch area parameter (beta _i). (b) pair-wise connectivity for the homogeneous patch case with (beta _i=0.5) (forall) i, versus normalized inter-patch distances (dfrac{D_{ij}}{L}) (forall) (i ne j).
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In the following section, we start by discussing some important characteristics of NPC. In “Homogeneous patches” section, we look at the system behavior for homogeneous patch parameters and contrast between different connectivity modes, namely (1) all-to-all connected, (2) fixed NBM, and (3) rewiring NBM. We study the homogeneous patch case first to focus on the influence of NPC on system dynamics without any confounding effects of patch heterogeneity. In “Heterogeneous patches” section, we discuss some general results for NBMs with heterogeneous patches, expressed through different patch carrying capacities.
NPC characteristics
For the case of (N=20) patches, individual patch (C_i) and pair-wise patch (C_{ij}) connectivity estimates are shown as functions of patch areas (beta _i) and inter-patch distances (D_{ij}), in Fig. 2a,b respectively. We observe a nearly linear growth in (C_i) as a function of patch areas in Fig. 2a, and a nonlinear (exponential) decrease in (C_{ij}) as a function of inter-patch distances in Fig. 2b. These results nicely illustrate the essential features of the NPC as formulated in Eq. (3): (a) larger patches have more incoming/outgoing connections as compared to smaller patches, and (b) closely located patches connect more frequently as compared to distant ones.
Homogeneous patches
The influence of dispersal rate d on species persistence are investigated and compared for the (1) all-to-all connected model, (2) fixed NBM, and (3) rewiring NBM. For uniformity in comparison, we consider the dispersal efficiency parameter (delta =1) in the following “Probability of persistence” and “Influence of patch density and network rewiring rates on persistence” sections.
Probability of persistence
Figure 3
Persistence probabilities (P_{mathrm{per}}) as a function of dispersal rate d, for different connectivity configurations with (N=20) and dispersal efficiency (delta =1). Curves: all-to-all connected (black), fixed NBM (green), and rewiring NBM (red) (rewiring every 100 time units). The ensemble sizes (number of network realizations) for these calculations are fixed at (N_{text{ensemble}}=100) for all three cases. Standard error values (mathrm{SE}(P_{mathrm{per}})), for NBM calculations are highlighted by error bars every four data points to avoid graph overcrowding. Corresponding biomass calculations can be seen in the Online Appendix Fig. A.1.
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Figure 3 highlights the influence of NPC on the system dynamics for the (N=20) case. For the (1) deterministic all-to-all connected case, species persistence (P_{{rm per}}=1) in the entire d range, and is mathematically expected to stay at the same value for arbitrarily high (d rightarrow infty) values—which is extremely counter intuitive considering the physical bounds and limitations in natural systems. Notably for an all-to-all connected system with (delta =1), the second term in Eq. (1) vanishes due to the parameter symmetry and the diffusive mean field nature of the term. Consequently, the dynamics of this system is essentially independent of the dispersal rate d, and therefore not affected by any changes in d. Hence in this case, local populations in each patch independently follow their logistic growth, and settle on the respective carrying capacity (K_i = K) (forall) i—which constitutes the stable equilibrium for the system. On the contrary, for the other two cases of (2) fixed and (3) rewiring NBMs, we observe decreasing (P_{mathrm{per}}) estimates for increasing d values. These results suggest that several network realizations in the NPC ensembles can lead to species extinction with higher dispersal rates in the NBMs. Furthermore, for rewiring NBM, (P_{mathrm{per}}) approaches very low values for higher d, before exhibiting species extinction (P_{mathrm{per}}=0) for (d approx 28) (not shown). On the contrary, we do not observe species extinction (P_{mathrm{per}}=0) for quite high d values for the fixed NBM case. Please see Online Appendix: Fig. A.1 for the corresponding average biomass calculations.
Influence of patch density and network rewiring rates on persistence
Figure 4
Persistence probabilities (P_{mathrm{per}}) as a function of the dispersal rate d for (a) fixed, and (b) rewiring (every 100 time units) NBMs. (N_{text{ensemble}}=100) network realizations were used for the simulations, for (N=10) (blue), (N=20) (red) and (N=40) (black) network sizes. Standard error values (mathrm{SE}(P_{mathrm{per}})) for these calculations are highlighted by error bars every four data points. Related biomass calculations can be seen in Online Appendix Fig. A.2.
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Figure 4 shows the relationship between species persistence probability (P_{mathrm{per}}), and dispersal rate d for fixed and rewiring NBMs. Since the value of largest possible distance in the system (L) is fixed (fixed landscape) for all calculations, an increase in number of patches N corresponds to an increase in patch density in the meta-population. The results suggest that a higher patch density supports species persistence in the system for higher dispersal rates d. For (N=10), (P_{mathrm{per}} > 0) (non-zero) for the fixed NBM case over the entire investigated range of d, extending to even higher d values (not shown here). This implies that there are always some network realizations in the ensemble that support species persistence. In contrast, the species go extinct at (d approx 5) for the rewiring NBM case. For (N=20), both fixed and rewiring NBMs yield non-zero (P_{mathrm{per}}) estimates in the investigated d range. However we observe a steeper decrease in persistence with increasing d, and consequently, extinction for (d approx 28) (not shown) for the rewiring NBM. For higher number of patches: (N=40), (P_{mathrm{per}} approx 1) for the entire d range for both fixed and rewiring NBMs, and this behavior extends to even higher d values above the range shown in Fig. 4. Our calculations suggest an exponential growth in the total number of connections as a function of number of patches N for a fixed landscape, and hence for increasing patch densities (see Online Appendix B, Fig. B.1). Therefore, we can conclude that higher patch densities tend to shift the (P_{mathrm{per}}) behavior closer to a highly connected case, which yields (P_{mathrm{per}}) results similar to the all-to-all connected situation. Overall, (N=20) seems to provide an appropriate trade-off between a system which is neither too sparse, nor a densely filled landscape, where a sparse system might lead to a higher proportion of isolated patches, and a dense system can mask the effects of spatially explicit connectivity. Therefore, we use (N=20) as the standard meta-population size in most of our calculations for the given set of parameter values.
For the case of rewiring NBMs, persistence probability (P_{mathrm{per}}) is also influenced by the rewiring time intervals of the connectivity matrix. Results in Fig. 5 compare the (P_{mathrm{per}}) estimates for fixed, and rewiring NBMs with different rewiring rates—every 10 and 100 time units. These results show that a faster rewiring NBM promotes species persistence in comparison to the fixed and slower rewiring NBMs, for a range of dispersal rates (d in (5,10))—a faster rewiring provides higher (P_{mathrm{per}}) estimates in this d range. However, the average biomass estimates for all these three cases are quite similar in this dispersal rate range (Online Appendix: Fig. A.3). However, for higher d values, species persistence for both rewiring NBMs is lower compared to the fixed NBM. Eventually, the rewiring NBMs exhibit species extinction for higher d, whereas, the fixed NBM still yields non-vanishing (P_{mathrm{per}}) estimates (not shown).
Figure 5
(P_{mathrm{per}}) estimates for (N=20), as a function of dispersal rate d, and (mathrm{SE}(P_{mathrm{per}})) are highlighted by error bars every four data points, for identical patches with fixed (black) versus different rewiring rates—every 10 (green) and 100 (red) time units. (N_{text{ensemble}}=100) network realizations were used for these calculations. Corresponding biomass estimates are provided in Online Appendix Fig. A.3.
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To better understand the mechanism of species extinction in the meta-population, we need to focus on the critical role of the between-patch interaction (second) term in Eq. (1), i.e. (-dleft( x_i-dfrac{delta _i}{k_{mathrm{in}}^{i}}sum nolimits _{j=1}^{N} A_{ij} x_jright)). As mentioned before, this term assumes that the dispersal process is diffusive in nature, and the diffusion occurs along a gradient from higher to lower values. With this assumption, the following situations can occur for any (d >0): (1) average input from other patches is higher than the patch population, i.e. (dfrac{delta _i}{k_{mathrm{in}}^{i}}sum nolimits _{j=1}^{N} A_{ij} x_j > x_i) (implies) (dleft( dfrac{delta _i}{k_{mathrm{in}}^{i}}sum nolimits _{j=1}^{N} A_{ij} x_j – x_iright) >0), the interaction term is positive implying an increase in the number of individuals within the patch via dispersal, i.e. net species movement is directed into the patch due to the population gradient, (2) average input from other patches is less than the patch population, i.e. (x_i >dfrac{delta _i}{k_{mathrm{in}}^{i}}sum nolimits _{j=1}^{N} A_{ij} x_j) (implies) (dleft( dfrac{delta _i}{k_{mathrm{in}}^{i}}sum nolimits _{j=1}^{N} A_{ij} x_j – x_iright) 0), the underlying NPC has a strong influence on the observed dynamics. The underlying connectivity matrix can lead to a situation where some patches have no incoming connections—such a situation is more likely in a sparsely connected network and/or for a network with low patch density. Absence of any incoming connections will lead to case (3) as discussed above, implying to net loss in the local biomass. Consequently, local populations will go extinct in these source-only patches once the dispersal rate is higher than the species growth rate for case (3). This extinction will decrease the biomass flux from these patches to the connected sink patches. With increasing dispersal rates, this will lead to case (2) for these sink patches and eventually they will also experience local extinction, thereby, enabling a cascade which leads to the extinction of the entire meta-population. In terms of bifurcation analysis, this extinction corresponds to a transcritical bifurcation (stability exchange) between equilibria with species persistence and extinction. An important point to consider is that for a fixed landscape, different NPC realizations can give rise to completely different persistence/extinction scenarios. Considering an NPC realization where all patches have at least one incoming connection, and another realization where one/some patches have no incoming connections (source-only case) or are isolated, the described mechanism indicates that species can persist for comparatively higher d values in the former case, as compared to the latter.
Figure 6
(i) (P_{mathrm{per}}) projection on the ((d,delta )) plane for homogeneous patches, and (ii) for the heterogeneous patch case. [(a.i),(a.ii)] correspond to all-to-all connected system, [(b.i),(b.ii)] to fixed NBM, and [(c.i),(c.ii)] to rewiring NBM with a rewire every 100 time units. The blue shaded area corresponds to meta-population extinction, i.e. (P_{mathrm{per}}=0), whereas red regimes correspond to (P_{mathrm{per}}=1). The boundary between persistence and extinction for the all-to-all connected system is indicated by a yellow curve which for comparison, is indicated in all panels. (N_{text{ensemble}}=100) for these calculations. Corresponding biomass calculations are shown in Online Appendix Fig. A.4.
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Figure 7
(i) (mathrm{SE}(P_{mathrm{per}})) projection on the ((d,delta )) plane for homogeneous patches, and (ii) for the heterogeneous patch case. [(a.i),(a.ii)] correspond to all-to-all connected system, [(b.i),(b.ii)] to fixed NBM, and [(c.i),(c.ii)] to NBM rewiring every 100 time units. The blue shaded area corresponds to regimes where the error is zero. Positive values are highlighted by colors as per the attached color bar. The boundary between persistence and extinction for the all-to-all connected system is again indicated by a yellow curve for comparison in all the panels. (N_{text{ensemble}}=100) for these calculations. The dynamical differences between all-to-all connected, and NBMs are even more obvious in these calculations.
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Dependence of persistence patterns on dispersal rate and efficiency
So far, the results were presented for cases with a dispersal efficiency (delta _i = delta = 1). However, an assumption of 100% efficiency of dispersal is unrealistic. Losses during the dispersal process are likely to occur in natural systems and there is a chance of the species not establishing after arrival in a new patch. To take these losses into account, we introduced (delta) in our system Eq. (1). In the following, we investigate species persistence for different connectivity scenarios for (delta in [0,1]), and (d in [0,20]). (P_{mathrm{per}}) ((mathrm{SE}(P_{mathrm{per}}))) behaviors for the homogeneous case are shown in the top row of Fig. 6 (Fig. 7), for (a.i) all-to-all connected, (b.i) fixed NBM, and (c.i) rewiring NBM, respectively.
Starting with the all-to-all connected case, we do not observe any transitions to (P_{mathrm{per}}=0) for (delta =1). The reason is related to parameter symmetries leading to a vanishing interaction term in Eq. (1) as discussed before. Therefore, the species in their respective patches can grow to the patch carrying capacities (K_i = K) (forall) i—see also the average biomass estimates in Online Appendix Fig. A.4. On considering dispersal efficiency (delta < 1), the interaction term in Eq. (1) does not vanish, and therefore, species extinction can arise, as seen in Fig. 6(a.i), where red (blue) areas correspond to persistence probabilities of one (zero). Linear stability analysis of the extinction equilibrium, (mathbf{x} ^*=mathbf{0} implies (x_1^*=0,x_2^*=0,ldots , x_N^*=0)^T), and any general (N ge 2) yields the following eigenspectrum, $$begin{aligned} begin{aligned} lambda _i={left{ begin{array}{ll}(r-d) + ddelta , &{}quad i=1 (largest) \ (r-d)-dfrac{ddelta }{(N-1)} &{}quad i=2,ldots ,N. end{array}right. } end{aligned} end{aligned}$$
(4) Note that these eigenvalues (lambda _i)s are independent of patch carrying capacities, therefore the persistence–extinction boundary is unaffected by parameter mismatch in the carrying capacities. Using the largest eigenvalue (lambda _1), one can follow the transition boundary of the transcritical bifurcation between persistence and extinction in the ((d,delta )) plane, which satisfies the expression (lambda _1 = (r-d) + ddelta =0). This boundary (yellow curve) is highlighted in Fig. 6(a.i) and is in excellent agreement with the theoretical estimates of this transition boundary (see Online Appendix Fig. A.4 for biomass calculations). Additionally, corresponding (mathrm{SE}(P_{mathrm{per}})) calculations in Fig. 7(a.i) show that the error values for the homogeneous all-to-all connected case are uniformly vanishing in the entire considered ((d,delta )) plane. This is due to the behavior of (P_{mathrm{per}}) which is 1 before the bifurcation boundary and 0 after the bifurcation leading to meta-population extinction.
For the spatially explicit Fig. 6(b.i) fixed NBM, and Fig. 6(c.i) rewiring NBM cases, we observe that the persistence–extinction transition boundary is not as sharp as for the all-to-all connected case. For comparison, the transition boundary for the all-to-all case is indicated by a yellow curve in these figures. For the fixed NBM case, starting from lower (delta) and d values, the transition becomes more uncertain as we increase the value of (delta)—which can be seen by the presence of lighter colored regimes corresponding to low, non-zero persistence probabilities. This ambiguity between persistence and extinction increases even further for higher (delta). These results suggest that for high (delta), and d values, species extinction is possible for the fixed NBM case contrary to the all-to-all connected network, where extinction is impossible in the similar parameter regime. The reasoning behind these results is quite straightforward considering how (P_{mathrm{per}}) is estimated. For high (delta) and d values, there is a proportion of connectivity configurations which lead to species extinction following the mechanism discussed in “Influence of patch density and network rewiring rates on persistence” section. Due to these configurations, we obtain (0< P_{mathrm{per}} < 1) in this case. This effect is even more pronounced for the rewiring network, where the blue (extinction) regimes extend to even higher (delta) and d values, thereby comparatively reducing the regimes of persistence in the parameter space. These results are quite contrary to the all-to-all connected system where the persistence is ensured with (P_{mathrm{per}} = 1) along the entire range of d values for high (delta). The dynamical differences between the all-to-all connected and NBMs are even more conspicuous in the corresponding (mathrm{SE}(P_{mathrm{per}})) calculations, see Fig. 7(b.i),(c.i). For the parameter regimes where (P_{mathrm{per}} = 1 (0)), the (mathrm{SE}(P_{mathrm{per}})) values are either zero or very small. For (0< P_{mathrm{per}} < 1), (mathrm{SE}(P_{mathrm{per}})) exhibit higher values implying a higher variability in the (P_{mathrm{per}}) estimates for NBMs. This comparison also highlights the differences between the fixed and rewiring NBMs. In correspondence to (P_{mathrm{per}}) estimates, fixed NBM results exhibit a higher variability for high (delta) and high d values, as compared to the rewiring NBM case. This is due to the fact that in these ranges of high variability for fixed NBM, the rewiring NBM predicts extinction of the meta-population.
It is quite natural that a meta-population with a local finite growth rate r and a biomass loss during dispersal, cannot sustain a population for ever increasing dispersal rates. For lower (delta) regimes, all three connectivity scenarios follow this reasoning. For higher (delta), the all-to-all connected system can sustain the meta-population for arbitrary high (d rightarrow infty) values with (P_{mathrm{per}} = 1). In comparison, our NBM implementations yield reasonable estimates of (P_{mathrm{per}} < 1) in high (delta) and d ranges. Additionally, rewiring NBMs show a higher likelihood of extinction than the fixed NBM—which highlights the fact that, everything else being constant, it is highly likely for a meta-population to exhibit extinction depending on the changes in underlying connectivity alone, which here, correspond to different NPC realizations.
Heterogeneous patches
Results for the heterogeneous patch case are shown in the bottom row of Fig. 6 for (a.ii) all-to-all connected, (b.ii) fixed NBM, and (c.ii) rewiring NBM. Here we investigate these three configurations for different patch areas (beta _i) (in) [0.3, 0.7], and consequently different patch carrying capacities, (K_i in [K_{min },K_{max }]), assigned to the patches in increments of (left( K_{max }-K_{min }right) /N). For our calculations in Fig. 6(a.ii),(b.ii),(c.ii), we chose (K_{min }=1.5), (K_{max }=3.5), and (N=20). We observe that the results for heterogeneous patches are quite similar to the results with identical carrying capacities. The similarity in the transition boundary can be explained by looking at the eigenspectrum in Eq. (4). The eigenspectrum for the all-to-all connected system is independent of the patch carrying capacities for the extinction equilibrium and therefore, the extinction threshold is not affected by the dissimilarity in the carrying capacities. Accordingly, for the all-to-all connected cases in Fig. 6(a.i),(a.ii), there are no differences, and (P_{mathrm{per}}) behaves identically in both the homogeneous [Fig. 6(a.i)] and heterogeneous patch [Fig. 6(a.ii)] cases. Like for the homogeneous case, (mathrm{SE}(P_{mathrm{per}})) calculations in Fig. 7(a.ii) again show that the error values for the heterogeneous all-to-all connected case are uniformly vanishing in the entire considered ((d,delta )) plane. For the fixed NPC case [Fig. 6(b.i),(b.ii)], we observe some differences for the transition boundaries for higher (delta) values. The (P_{mathrm{per}}) estimates at the transition boundaries for high (delta) and d values are lower (lighter blue dots) in the heterogeneous case [Fig. 6(b.ii)] when compared to the homogeneous case [Fig. 6(b.i)], thereby signifying that in the heterogeneous case, more realizations in the ensemble close to the transition lead to extinction. At the same time, we observe that patch heterogeneity shifts the extinction threshold towards higher (delta) values, as compared to the homogeneous case. This implies that for the case of heterogeneous patches, we will observe species extinction for higher (delta) values where the homogeneous system still supports persistence. A similar pattern can be observed for the homogeneous [Fig. 6(c.i)] and heterogeneous [Fig. 6(c.ii)] rewiring SEMs. Similar to the homogeneous case, dynamical differences between all-to-all and SEMs are yet again more obvious in (mathrm{SE}(P_{mathrm{per}})) calculations in Fig. 7(b.ii),(c.ii). These observations suggest that unlike in the all-to-all connected case, patch heterogeneity appears to play an essential role in determining the extinction threshold for meta-populations as a function of d and (delta) in fixed, as well as rewiring NBMs. More
