# The role of spatial scale in organism–environment positive feedback

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## Abstract

Organism–environment positive feedback (i.e., ecosystem self-modification or facilitation) will incur bistability, which is often disadvantageous to biological conservation and ecosystem restoration. Using a spatially correlated equation based on pair approximation and simulation, we found that the feature of bistability in the positive feedback system strongly depends on spatial scale of organism–environment feedback. It will mitigate and even disappear when the interaction between organisms and environment is localized spatially, while the population will lessen globally when dispersal colonization is limited. As the spatially local influence is a basic property of real ecological systems (especially for sessile organisms), but classical ecological models based on mean-field assumption (or mass action) does not consider the impact of spatial scale. This implies that spatial configuration and local facilitation could be essential process for the stability and maintenance of ecosystem.

## Keywords

Self-modification Bistability Spatial structure Pair approximation Simulation Suitable patches Neighborhood## 1 Introduction

Both empirical and theoretical studies have revealed that the organism–environment positive feedback can incur alternative stable states, meaning that there are two attracting states for the system (i.e., bistability) [7, 8, 9, 10, 11, 12, 13], which could account for the catastrophic ecosystem shifts [8, 14, 15, 16]. Although external conditions (e.g., climate, nutrient loading, biotic exploitation, or habitat fragmentation) change gradually, the ecosystem may shift abruptly [17]. In fact, many empirical studies have proved the appearance of catastrophic shift in communities. For example, shallow lakes transparency was suddenly lost and became turbid [18, 19]; reef ecosystems emerged phase shift because corals were overgrown caused by fleshy macroalgae [20, 21]; savannahs had a sudden drama shift due to bushes encroachment [22, 23]. Obviously, the catastrophic shifts are much more disadvantageous to conservation and restoration of the ecosystem [19, 24, 25, 26] and must be considered in policy provision and management [27, 28].

Because the availability and spatial distribution of suitable habitats significantly affect population persistence through changing local spatial clustering [29, 30, 31, 32, 33, 34], the spatial distribution and dynamics of the population could play an important role in population persistence and ecosystem conservation [35, 36, 37]. However, few studies are concerned with the effect of spatial scale on alternative stable states and catastrophic shifts [38]. It is necessary to incorporate spatial scale into theoretical models in studying positive ecological feedback [39, 40, 41, 42]. We here consider the scale-dependent positive feedback and address this issue through a spatially correlated model based on pair approximation and simulation [43, 44, 45]. We find that spatially limited interactions between external environment and organism are able to mitigate ecosystem bistability. This implies that spatial configuration and local facilitation could be an essential process for the stability and maintenance of population.

## 2 Models

### 2.1 Mean-field model

*h*is the probability that a randomly chosen patch is suitable (also explained as the density of suitable patches), and

*p*the probability that a randomly chosen patch is occupied by focal organism (also explained as the density of occupied patches). Parameter

*c*,

*e*, and

*d*represent colonization rate to empty suitable patches, extinction rate of local occupancy, and habitat destruction rate. The parameter \(\lambda \) and \(\mu \) are habitat restoration rates respectively by organism positive feedback (e.g., self-modification) and other external factors. The parameters are all nonnegative. Notably, the model (Eq. 1) only approximates the dynamical behavior of patchy states at regional scale, but does not catch spatial structure at local scale [13]. In order to study spatial-scale effects, we considered spatially local interaction by pair approximation method and simulation based on cellular automata in following two subsections [46, 47].

### 2.2 Pair approximation model

*ij*and

*ji*are no difference where \(i, j = -, 0, +)\), and their dynamics can be given by the following ordinary differential equations [44, 47]:

*ij*, \(q_{k\left| {ij} \right. } \) is the conditional probability that a random chosen neighboring patch of

*i*-patch in

*ij*-patch pair is in state

*k*. The parameter \(n_{1 }\) and \(n_{2}\) are the neighborhood sizes of colonization and restoration habitat of organism [48]. The former describes the colonization ability of the organism, and the later shows its influence range on external environment. If letting \(p_{i}\) represent the probability that a randomly chosen patch is in state

*i*, in term of probability theory, we have the following equalities: \(p_{--} +p_{-0} +p_{-+} =p_- \), \(p_{0-} +p_{00} +p_{0+} =p_0 \), \(p_{+-} +p_{+0} +p_{++} =p_+ \), and \(p_- +p_0 +p_+ =1\). We must indicate that Eq. 2 is not closed because the dynamics of the patch pairs depend on patch triplets since \(q_{k\left| {ij} \right. } =p_{kij} /p_{ij} \). In this case, a generally used approximation method (i.e., pair approximation) is to assume the probability of finding a

*k*-patch in the neighborhood of

*i*-patch does not depend on its other neighbors (i.e., \(q_{k\left| {ij} \right. } =q_{k\left| i \right. } )\), and then Eq. 2 is rewritten as

Transition rates in simulation

State transition | Transition rate |
---|---|

+ \(\rightarrow \) 0 | |

+ \(\rightarrow \) − | |

0 \(\rightarrow \) − | |

0 \(\rightarrow \) + | \(c{\bar{p}}_1 \) |

\(- \rightarrow \) 0 | \(\lambda {\bar{p}}_2 +\mu \) |

In the pair approximation model, the three types of patches (\(-, 0\) and \(+\)) are discrete. We cannot simply distinguish these three states in the ecosystem because there are still transitional states between these three states in reality. This is the limitation of the pair approximation model.

### 2.3 Cellular automata simulation

Furthermore, a cellular automata model can be built to simulate the spatial positive feedback system on lattice space. We here identify five possible types of patchy state transition (Table 1): occupied patches will extinct (+\(\rightarrow \)0) at rate *e* or be destroyed \((+\rightarrow -)\) at rate *d*; empty suitable patches will also be destroyed \((0\rightarrow -)\) at rate *d* or be reoccupied (0\(\rightarrow \)+) at rate \(c{\bar{p}}_1 \) where \({\bar{p}}_1 \) is the fraction of occupied patches in its neighborhoods; and unsuitable patches will be modified and become suitable \((-\rightarrow 0)\) at rate \(\lambda {\bar{p}}_2 +\mu \) where \({\bar{p}}_2 \) is the fraction of occupied patches in its neighborhoods. We adopted periodic boundary condition and asynchronous update in our simulations [49].

## 3 Results

*c*,

*e*,

*d*, \(\lambda \) and \(\mu \) becomes small with decreasing restoration neighborhood size \(n_{2}\) (Fig. 5).

In summary, locally confined organism–environment interaction and globally colonization could mitigate and even eliminate bistable phenomenon caused by positive feedback, and thus be beneficial for the persistence and restoration of ecosystem instead of catastrophic shift. This result implies that the spatial scale of organism–environment feedback plays a non-negligible role in ecosystem.

## 4 Discussion

In recent years, evidence of studies suggests that positive feedback play crucial roles in many communities [51], and alternative stable states (discontinuous transition) often exist in ecosystems [25, 52]. More empirical evidences indicated the importance of positive feedback loop for alternative stable states. For example, there are two alternative stable states in drylands: vegetated and bare soil. More vegetation can improve the infiltration and retention of water and help to establish better growth conditions. On the contrary, less vegetation makes the soil lack the physical protection of vegetation, which leads to soil erosion and crusting. Thus, worse growth conditions are established and bare soil is easy to form [53, 54]. Another classic example is the alternative stable states of clarity and turbidity in shallow lakes. Many submerged plants can absorb more nutrients through photosynthesis, so phytoplankton decreases and the water becomes clear. Conversely, when the number of submerged plants is less, the water becomes turbid because nutrient loading is external [19]. Moreover, many theoretical studies also already revealed that if the organism–environment positive feedback are strong enough, alternative stable states are possible [13, 40, 46, 55, 56, 57], including the mean-field model and the model with spatial structure.

Though the positive feedback can incur alternative stable states, but there has been much work trying to use positive feedback to restore systems with alternative stable states [58, 59]. For example, in shallow lakes, the number of Zooplankton increases through reducing the abundance of benthic fishes. Since Zooplankton predates on phytoplankton, this reduces the turbidity of water and increase the light for submerged plants [60]. In degraded semiarid ecosystems, original shrublands and forests may be restored by ‘nurse plant’ [61]. And theoretical models of alternative stable states that incorporate system positive feedbacks are now being applied to the dynamics of recovery in degraded systems [59].

In this paper, the scale-dependent positive feedback system, which presented by Rietkerk et al. [38], is considered. We mainly discuss the effect of scale-dependent positive feedback on the bistability (discontinuous transition) in organism–environment. Furthermore, Kéfi et al. have also raised the question of when positive feedback may lead to discontinuous transition [55]. So it is particularly important for early warning of this discontinuous transition [62]. Since positive feedback is an element of the threshold (tipping point) of discontinuous transition, the transition from one state to another is very sudden at the threshold (tipping point). If the threshold (tipping point) of the complex system can be found, it is possible that the ecosystem will avoid this discontinuous transition and have the opportunity to change in a positive direction [63].

In addition, we find these thresholds (tipping points) for the different models (i.e., the pair approximation model and simulation) are different (Fig. 6). We cannot accurately find this threshold (tipping point) to alert discontinuous transition because simulation has randomness. In pair approximation model, we can find this threshold (tipping point) more accurately and provide early warning for the system, thus providing better guidance for the management of the ecosystem. However, under the assumption of mean field, this system have the characteristics of homogeneity and high connectivity, which provides resistance to the alternative states transition of the system until the system reaches the threshold (tipping point) [64]. Therefore, although this threshold (tipping point) is lowered, it is possible to provide an erroneous early warning. This is because the spatial structure is ignored in the mean-field model, so that it does not conform to the real world system.

Since the effect of the neighborhood sizes of colonization and restoration habitat of organism on the population is just the opposite. Therefore, when the impact of individual organisms on the environment is restored at fine scale (i.e., small \(n_{2})\), and the individual must disperse as much as possible (i.e., large \(n_{1})\), the system must be able to resist the bistable phenomenon. In addition, bistability also depends on the initial value [65]. Population with lower density is less prone to disperse, but population with higher density tends to disperse [56]. If the initial value of population is large, the bistable phenomenon may be avoided.

In fact, positive feedback among multiple organisms is more common in ecosystems. In a more complex system, the influence of this spatial-scale-dependent positive feedback on populations is worth further discussion and research. And the integration of ecological factors in research is very important, which also provides potential interest and direction for future theoretical researchers.

## Notes

### Acknowledgements

We are grateful to two anonymous reviewers for their helpful comments. This work is supported by National Natural Science Foundation of China (31360104), Sheng Tongsheng Innovation Funds (GSAU-STS-1540), and Special Funds for Discipline Construction of Gansu Agricultural University (GAU-XKJS-2018-143).

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