# Temporal Resource Continuity Increases Predator Abundance in a Metapopulation Model: Insights for Conservation and Biocontrol

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Landscapes

_{i}= 1), which offers relatively abundant basal food resources for prey populations, or a low-resource state (x

_{i}= 0), which offers relatively few resources for prey. We thus defined three patch types: (1) early-season high-resource (e.g., early-season crop); (2) late-season high-resource (e.g., late-season crop); or (3) continuously low-resource (e.g., highly disturbed or developed area). Because our focus in this study is investigating the potential role of cropland as habitat and simulating dynamics of predators and prey in managed agricultural landscapes, we intentionally exclude a fourth potential resource state, one of continuously high-resources (e.g., diverse natural habitats that have resources early and late in the season). By varying the number and location of high-resource patches in each half of the growing season, we can generate model landscapes with different combinations of basal resource amount and temporal variance (Figure 1). We quantify the total, or season-long, amount of basal resources R in a landscape as the sum of resource state values in the first and second half (A and B) of the growing season:

_{R}) to standardize for the level of R, where

_{R}values indicate landscapes where resources are synchronously available and concentrated in only half of the growing season, which creates a “gap”, or period of low resource availability, during the other half of the season (Figure 1A,C). In contrast, low CV

_{R}values indicate landscapes where resources are asynchronously available and more evenly distributed throughout the growing season (Figure 1B,D).

#### 2.2. Predator–Prey Metapopulation Model

_{i,t}is the density of prey in patch i at time t; r is the maximum per capita rate of growth; and I

_{i,tG}and E

_{i,tG}represent prey immigration and emigration, respectively, to and from patch i at time t. Parameter definitions and values are summarized in Table A1 (Appendix A). Prey carrying capacity K depends on the resource state of patch i at time t. We assumed K = 100 in high-resource patch states (x

_{i}= 1) and K = 10 in low-resource patch states, (x

_{i}= 0). Prey have both a constant background mortality at rate m

_{G}, and mortality from predation in each patch based on a per capita attack rate a. For the set of analyses involving two specialist prey (S1 and S2), we also used Equation (4) to model their population dynamics, substituting S1 or S2 for G. For simplicity, we assumed a type I functional response for both generalist and specialist prey, but also explored cases where predation followed a type II functional response [19] and found qualitatively similar results (not shown). The initial density of G, S1, and S2 in each patch was set at 1. Predator and prey subpopulations that fell below a density of 1 × 10

^{−6}are assumed to be locally extinct, but can be recolonized by immigration from another patch.

_{i,t}is the density of predators in patch i at time t, and I and E represent predator immigration and emigration, respectively, to and from patch i at time t. Predators suffer constant mortality at rate m

_{P}. We used a similar equation to describe predator subpopulation dynamics in models involving two specialist prey:

^{−6}are assumed to be locally extinct, but can be recolonized by immigration from another patch.

_{G}= 0.2) to their subpopulation size. For each patch and time step, d

_{G}G

_{i,t}(or d

_{S}

_{1}S1

_{i,t}or d

_{S}

_{2}S2

_{i,t}) prey move in a randomly selected direction to a new patch. Dispersal distance is randomly drawn from a Poisson distribution (λ = 3). Thus, prey dispersal distance is usually within one to three patches, with larger distances increasingly less likely. To avoid edge effects, the model assumes a periodic boundary condition so that dispersers that move off the landscape in one direction emerge on the opposite side. Predator dispersal follows similar rules, but the fraction of each predator subpopulation dispersing d

_{P}is dependent on both predator and prey density. That is, d

_{P}= µ

_{G}+ µ

_{P}, where

_{G}and µ

_{P}approach their maximum and γ is the maximum rate of predator dispersal in response to predator density (i.e., maximum µ

_{2}). Therefore, d

_{P}approaches 1.0 as G

_{i,t}approaches 0 (negative prey density dependent dispersal) and d

_{P}approaches γ as P

_{t}increases (positive predator density-dependent dispersal). For models involving two specialist prey, d

_{P}= µ

_{S}

_{1S2}+ µ

_{P}, where

#### 2.3. Model Analysis

_{R}). Because the season-long resource amount had a strong effect on metapopulation densities (described below), we examined the effects of temporal variance in each of 100 landscapes with R fixed at 51, 102, 154, 205, and 256 (i.e., high-resource habitat in 20, 40, 60, 80, and 100% of patches). We evaluated R effects on metapopulations using the proportion of maximum R (pR

_{max}), which, for the 256-patch landscapes, equals R/256. Because predators respond directly to prey dynamics, rather than variability in R, we also calculated the temporal variance in prey metapopulation size (CV

_{prey}) over the final 100 time steps:

_{P–}prey

_{0})/prey

_{P}). This allowed us to examine how biological control varied with temporal variance and habitat amount.

## 3. Results

#### 3.1. Effects of Resource Amount and Continuity on Prey

_{R}) was also consequential for prey abundance, but this was only evident in high-resource landscapes and the direction of the relationship depended on prey specialization. In landscapes with more high-resource habitat patches (higher levels of pR

_{max}), generalist prey abundance increased as temporal variance decreased (i.e., with greater resource continuity; Figure 2C). The opposite relationship was true in landscapes with two specialist prey, whose summed abundance decreased with lower temporal variance (Figure 2D).

#### 3.2. Effects of Resource Amount and Continuity on Predators

_{max}≈ 0.4), but then decreased (Figure 3B, red points). As with prey, the effects of temporal resource patterns on predator populations were most apparent in high-resource landscapes. Predator abundance generally decreased with greater temporal variance, especially when feeding on specialists (Figure 3C,D).

_{prey}; Figure 4) and the direct responses of predators to this prey variance (Figure 5). For both the single generalist and two specialist prey populations, the temporal variability of prey abundance increased with the total resource amount even when basal resources were continuous (i.e., CV

_{R}= 0; Figure 4, blue points); basal resource discontinuity exacerbated the effect (Figure 4, red points). Thus, as landscapes filled with more available high-resource patches, and/or when basal resources were available more temporally variable, prey populations cycled with greater amplitude. This is apparent in, for example, metapopulation dynamics, illustrating how both the resource amount (pR

_{max}) and resource temporal variance (CV

_{R}) increase prey temporal variance (CV

_{prey}) (Figure A2 and Figure A3). The consequence of this high variability in prey abundance was a relative decrease in predator abundance across all levels of basal resource amount (Figure 5).

#### 3.3. Prey Suppression

## 4. Discussion

#### 4.1. Generalist Prey

#### 4.2. Specialist Prey

#### 4.3. Applications to Biological Control and Conservation

#### 4.4. Limitations and Future Directions

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Parameter | Definition | Value | |
---|---|---|---|

Landscape | R | Number of high-resource patches in the landscape | 0 to 256 |

pR_{max} | Proportion of maximum R | 0 to 1 | |

CV_{R} | Coefficient of variation of R between seasons | Equation (3) | |

Prey | G_{it} | Generalist prey population density in patch i at time t | Equation (1) |

S1_{it} | Specialist 1 prey population density in patch i at time t | Equation (1) | |

S2_{it} | Specialist 2 prey population density in patch i at time t | Equation (1) | |

S_{it} | Combined specialist prey density in patch i at time t | S2_{it} + S2_{it} | |

r | Prey intrinsic growth rate | 1.0 | |

K_{it} | Prey carrying capacity in patch i at time t | 100 (x_{i} = 1) or 10 (x_{i} = 0) | |

m_{G, S}_{1, or S2} | Prey mortality rate | 0.1 or 0.8 | |

d_{G, S}_{1, or S2} | Proportion of G, S1, or S2 dispersing from each patch | 0.2 | |

I_{it} | Sum of prey immigrating to patch i at time t | - | |

E_{it} | Prey emigrating from patch i at time t | - | |

CV_{prey} | Prey (G or S) within-season temporal variance (coefficient of variation) | Equation (10) | |

Predator | P_{it} | Predator population density in patch i at time t | Equations (5) and (6) |

c | Predator conversion efficiency of prey | 0.6 | |

a | Predator attack rate | 0.6 | |

m_{P} | Predator mortality rate | 0.1 | |

d_{P} | Proportion of P dispersing to new patch | Equations (7) and (8) | |

I_{Pit} | Sum of predators immigrating to patch i at time t | - | |

E_{Pit} | Predators emigrating from patch i at time t | - |

**Figure A1.**Model runs with predator dispersal set at a fixed rate of 0.2 showing the effects of proportion of high-resource habitat (pR

_{max}) and resource temporal variance (CV

_{R}, colored blue-to red) on (

**A**) abundance of a single generalist prey, (

**B**) summed abundance of two specialist prey, (

**C**) predators feeding on a single generalist, and (

**D**) predators feeding on two specialists.

**Figure A2.**Example of metapopulation dynamics of the generalist prey (G; green) and predator (P; blue). Landscapes modeled in panels (

**A**–

**D**) and associated values of R and CV

_{R}correspond to those in Figure 1A–D. For clarity, only the first 100 of 500 generations are shown.

**Figure A3.**Example of metapopulation dynamics of the specialist prey (G; green) and predator (P; blue). Landscapes modeled in panels (

**A**–

**D**) and associated values of R and CV

_{R}correspond to those in Figure 1A–D. For clarity, only the first 100 of 500 generations are shown.

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**Figure 1.**Conceptual diagrams (left) and corresponding model landscapes (right) for landscapes with different proportions of high-resource habitat (pR

_{max}) and temporal variance (CV

_{R}). Gray lines represent the amount of resources provided by early-season habitat (gray patches). Black lines represent the amount of resources provided by late-season habitat (black patches). White patches provide consistently few basal resources year-round. Panels (

**A**–

**D**) show model landscapes with different patterns of resource amount and continuity.

**Figure 2.**Prey response to the amount (pR

_{max}, top panels) and variability (CV

_{R}, bottom panels) of high-resource habitat in the landscape. Left panels (

**A**,

**C**) show the response of a single generalist prey, G. Right panels (

**B**,

**D**) show the sum of the density response of two specialist prey, S1 + S2. Points represent individual model runs with landscapes varying across parameter space. Labels (1A–1D) within panels C and D indicate model runs corresponding to the landscapes shown in Figure 1.

**Figure 3.**Predator response to the amount (pR

_{max}, top panels) and variability (CV

_{R}, bottom panels) of high-resource habitat in the landscape. Left panels (

**A**,

**C**) show the response of predators feeding on a single generalist prey. Right panels (

**B**,

**D**) show the response of predators feeding on two specialist prey. Points represent individual model runs with landscapes varying across parameter space. Labels (circles 1A–1D) within panels (

**C**,

**D**) indicate model runs corresponding to the landscapes shown in Figure 1.

**Figure 4.**Variability in the abundance of (

**A**) generalist and (

**B**) summed specialist prey changes with increasing proportion of high-resource habitat (pR

_{max}) in the landscape and greater temporal variance in R (CV

_{R}, color-coded blue to red).

**Figure 5.**Predator response to variation in prey (CV

_{prey}) when feeding on (

**A**) one generalist prey species G and (

**B**) two specialist prey S1 + S2. Colors indicate the proportion of patches with maximum resource amount (pR

_{max}), or the amount of the landscape providing high-resource habitat, for each of 100 separate model runs. Gray points represent a set of 1000 model runs where pR

_{max}was randomly selected between 0 and 1. Labels (circles 1A–1D) indicate model runs corresponding to the landscapes shown in Figure 1.

**Figure 6.**Suppression of (

**A**) generalist and (

**B**) specialist prey in response to changing temporal variance of basal resources (CV

_{R}). Colors indicate the proportion of patches maximum resource amount (pR

_{max}), or the amount of the landscape providing high-resource habitat, for each of 100 separate model runs. Gray points represent a set of 1000 model runs where pR

_{max}was randomly selected between 0 and 1. Labels (circles 1A–D) indicate model runs corresponding to the landscapes shown in Figure 1.

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**MDPI and ACS Style**

Spiesman, B.; Iuliano, B.; Gratton, C.
Temporal Resource Continuity Increases Predator Abundance in a Metapopulation Model: Insights for Conservation and Biocontrol. *Land* **2020**, *9*, 479.
https://doi.org/10.3390/land9120479

**AMA Style**

Spiesman B, Iuliano B, Gratton C.
Temporal Resource Continuity Increases Predator Abundance in a Metapopulation Model: Insights for Conservation and Biocontrol. *Land*. 2020; 9(12):479.
https://doi.org/10.3390/land9120479

**Chicago/Turabian Style**

Spiesman, Brian, Benjamin Iuliano, and Claudio Gratton.
2020. "Temporal Resource Continuity Increases Predator Abundance in a Metapopulation Model: Insights for Conservation and Biocontrol" *Land* 9, no. 12: 479.
https://doi.org/10.3390/land9120479