Philippa Kaur

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ModelWe model a tri-trophic food chain of one plant, one herbivore and one predator population on one or two habitat patches and complex meta-food-webs consisting of 10 plants and 30 animals in different landscapes containing 50 patches. The feeding dynamics are constant overall patches and are determined by the allometric food-web model by Schneider et al. 201633. We integrate dispersal as species-specific biomass flux between habitat patches according to Ryser et al. 201934. With the use of a dynamic bioenergetic model we formulate feeding and dispersal dynamics in terms of ordinary differential equations. The rate of change in biomass densities of a species are the sum of its biomass loss by metabolism, being preyed upon and emigration and its biomass gain by feeding and immigration. For detailed equations and for model parameters see section Equations and parameters and the supplement (Supplementary Table 1).Local food-web dynamicsFollowing the allometric food-web model by Schneider et al. 201633 each species is fully characterised by its average adult body mass. For the complex food-web log10 body masses were randomly drawn from a uniform distribution from 0 to 3 for plants and from 2 to 6 for animals. For the food chain the plant body mass was set to 102, the herbivore body mass to 104 and the predator body mass to 106. We set mass ratios of the herbivore to the plant and the predator to the herbivore to the optimum of 100, thus the respective resource being a one-hundredth of its consumer’s body mass. This simplifies feeding efficiency rates (see section Equations and parameters; Li,j, Eq. (5)) to 1 in the case of a food chain. Trophic dynamical parameters, such as metabolic rates and feeding rates, scale with body masses of model species. Also, we assume a type-II functional response for the food chain and a slight nonlinearity of the functional response in the food web as this stabilises persistence in more complex systems. Compared to Ryser et al 2019, capture rates were reduced to 5% to achieve viable food chains and food webs to increase the stability in the absence of interference competition.Nutrient modelWe have an underlying nutrient model with one nutrient that is driving the nutrient uptake and therefore the growth rate of the plant population11,33. The nutrient model consists of one nutrient, a nutrient turnover rate of 0.25 and a nutrient supply concentration. The nutrient supply concentration was varied to get eutrophic and oligotrophic patches (see Setup).Spatial dynamicsWe model dispersal between local communities as a dynamic process of emigration and immigration, assuming dispersal to occur at the same timescale as the local population dynamics35. Thus, biomass flows change dynamically between local populations and the dispersal dynamics directly influence local population dynamics and vice versa25.Dispersal rates of animals are modelled with an adaptive emigration rate depending on the net growth rate on the given patch. Dispersal ranges depend on the body masses of our model species with larger species having a higher dispersal range. We model a hostile matrix between habitat patches that does not allow feeding interactions to occur during dispersal. Depending on the scenario, we define a landscape with one, two or 50 patches. In cases with two or 50 patches, their locations are spatially explicit and were chosen in a way that the distances between reflect the dispersal loss of the predator across the matrix hostility gradient.Emigration and immigrationBased on empirical observations36 and previous theoretical frameworks13,22,37, we assume that the maximum dispersal distance of animal species increases with their body mass. For simplicity, we do not let the plants disperse, as they do not move themselves and the dispersal of plant propagules strongly depends on their dispersal strategy. We model emigration rates as a function of each species’ per capita net growth rate, which is summarising local conditions such as resource availability, predation pressure, and inter- and intra-specific competition25 (but see Sensitivity Analyses for dispersal models with constant dispersal or non-body-mass-scaled dispersal ranges). Dispersal losses scale linearly with the distance between two patches and are 100% in scenarios with only one patch or when the distance between the two patches surpasses the dispersal range of an animal. Even though we model dispersal losses according to dispersal distances, this loss term could also represent any other sort of dispersal loss. For numerical reasons, we did not allow dispersal flows smaller than 10−10.Numerical simulationsWe initialised each local population with a biomass density randomly sampled from a uniform probability density within the interval (0,10). Starting from these random initial conditions, we numerically simulated food web and dispersal dynamics over 100,000 time steps by integrating the system of differential equations implemented in C++ using procedures of the SUNDIALS CVODE solver version 2.7.0 (backward differentiation formula with absolute and relative error tolerances of 10−10) and the time series of biomass densities were saved for last 10,000 time steps. For numerical reasons, a local population was considered extinct and was set to 0 once its biomass density dropped below 10−20. Based on the empirically derived metabolic rates, these 100,000 time steps correspond to ~11 years. Our model does, however, not account for time spent for organisms’ other non-trophic activities such as sleeping or mating. Thus, the time scales of the simulation should only be compared with caution to natural time scales of population dynamics. Transient dynamics usually equilibrate within the first few thousand time steps.Equations and parametersOur model formulates the change of biomass densities over time in ordinary differential equations. Given the empirical origin of metabolic rates used in our model, one time step corresponds to an hour and body masses are in mg, areas of patches are not defined. The feeding links (i.e. who eats whom) are constant overall patches and are as well as the feeding dynamics determined by the allometric food-web model by Schneider et al. 201633. We integrate dispersal as species-specific biomass flow between habitat patches. Using ordinary differential equations to describe the feeding and dispersal dynamics, the rate of change in biomass density Bi,z of species i on patch z is given by$$frac{d{B}_{i,z}}{{dt}}[{mg}* {{{{{{{mathrm{Area}}}}}}}}^{-1}* {h}^{-1}]={B}_{i,z}mathop{sum}limits_{j}{e}_{j}{F}_{{ij},z}-mathop{sum}limits_{j}{{B}_{j,z}F}_{{ji},z}-{x}_{i}{B}_{i,z}-{E}_{i,z}+{I}_{i,z}({{{{{rm{for}}}}}}; {{{{{rm{animals}}}}}})$$
(1)
$$frac{d{B}_{i,z}}{{dt}}[{mg}* {{{{{{{mathrm{Area}}}}}}}}^{-1}* {h}^{-1}]={r}_{i}{G}_{i}{B}_{i,z}-mathop{sum}limits_{j}{B}_{j,z}{F}_{{ji},z}-{x}_{i}{B}_{i,z}({{{{{rm{for}}}}}}; {{{{{rm{plants}}}}}})$$
(2)
with the first three terms describing local trophic dynamics and the last two terms describing emigration, Ei,z (Eq. 9), and immigration, Ii,z (Eq. 11). For simplicity, we do not let plants disperse. Trophic dynamics are driven by following three processes. First, predation or herbivory on species j with assimilation efficiency e (ej = 0.545, if j is a plant, typical for herbivory; ej = 0.906 if j is an animal, typical for carnivory38) and the functional response Fij,z (Eq. 3) for animals, and a nutrient dependent growth (Eq. 7) for plants. Second, losses due to predation or herbivory, respectively. Third, losses by metabolic demands with xi = xAmi−0.305 with scaling constant xA = 0.141 (tenfold laboratory metabolic rate39 at a temperature of 20° Celsius to represent field metabolic rates) for animals and xi = xPmi−0.25 with xP = 0.138 for plants. We used a dynamic nutrient model (Eq. 8) as the energetic basis of our food web. Each species i is fully characterised by its average adult body mass mi. Body masses determine the interaction strengths of feeding links as well as the metabolic demands of species. Data from empirical feeding interactions are used to parametrise the functions that characterise the optimal prey body mass and the location and width of the feeding niche of a predator33. From each mi a unimodal attack kernel, called feeding efficiency Lij is constructed which determines the probability of consumer species i to attack and capture an encountered resource species j. We model Lij as an asymmetrical hump-shaped Ricker’s function (Eq. 5) that is maximised for an energetically optimal resource body mass (optimal consumer-resource body mass ratio Ropt = 100) and has a width of γ. The maximum of the feeding efficiency Lij equals 1. Supplementary table 1 is an overview of the standard parameter set for the equations. See also Schneider et al. 201633 for further information regarding the allometric food-web model.Functional response$${F}_{{ij},z}=frac{{omega }_{i}{b}_{i,j}{R}_{j,z}^{1+q}}{1+{omega }_{i}{sum }_{k}{b}_{{ik}}{h}_{{ik}}{R}_{k,z}^{1+q}}cdot frac{1}{{m}_{i}}$$
(3)
Per unit biomass feeding rate of consumer i as function of the biomass density of the resource Rj, with bi,j, resource-specific capture coefficient (Eq. 4); hi,j, resource-specific handling time (Eq. 6); ωi = 1/(number of resource species of i), an inefficiency parameter for generalists assuming that generalist are less adapted in for example search patterns or hunting strategies to a specific prey species; and q, the Hill coefficient for nonlinearities in density dependency (if q = 0 it is a Type-II functional response, if q = 1 it is a Type-III functional response).Capture coefficient$${b}_{{ij}}=f{a}_{k}{m}_{i}^{{beta }_{i}}{m}_{j}^{{beta }_{j}}{L}_{{ij}}$$
(4)
Resource-specific capture coefficient of consumer species i on resource species j scaling the feeding kernel Lij by a power function of consumer and resource body mass, assuming that the encounter rate between consumer and resource scales with their respective movement speed. This body mass scaling of encounter rates is assumed to occur before the attempt of a predator to capture its prey is made. We differentiate between carnivorous and herbivorous interactions with each comprising a constant scaling factor for their capture coefficients ak with k ∈ 0, 1 (a0 = 15 for carnivorous species and a1 = 3500 for herbivorous species). For plant resources, ({m}_{j}^{{beta }_{j}}) was replaced with the constant value of 1 (as plants do not move).Feeding efficiency$${L}_{i,j}={left(frac{{m}_{i}}{{m}_{j}{R}_{{{{{{{mathrm{opt}}}}}}}}}{e}^{1-frac{{m}_{i}}{{m}_{j}{R}_{{{{{{{mathrm{opt}}}}}}}}}}right)}^{gamma }$$
(5)
The probability of consumer i to attack and capture an encountered resource j (which can be either plant or animal), described by an asymmetrical hump-shaped curve (Ricker’s function), centered around an optimal consumer-resource body mass ratio Ropt = 10033 and with γ that that affects the width of the hump. An increase in γ results in a decrease in the width.Handling time$${h}_{{ij}}={h}_{0}{m}_{i}^{{eta }_{i}}{m}_{j}^{{eta }_{j}}$$
(6)
The time consumer i needs to kill, ingest, and digest resource species j, with scaling constant h0 = 0.4 and allometric exponents ηi = −0.48 and ηj = −0.6640.Growth factor for plants$${G}_{i}=frac{N}{{K}_{i}+N}$$
(7)
Species-specific growth factor of plants determined dynamically by the nutrient; with Ki, half-saturation densities determining the nutrient uptake efficiency assigned randomly for each plant species i and (uniform distribution within (0.1, 0.2)).Nutrient dynamics$$frac{d{N}_{z}}{{dt}}=Dleft(S-Nright)-mathop{sum}limits_{i,z}{r}_{i}{G}_{i}{P}_{i,z}$$
(8)
Rate of change of nutrient concentration N of nutrient on patch z, with global turnover rate D = 0.25, determining the rate at which nutrients are refreshed and the nutrient supply concentration S.Generating landscapesWe generated different fragmented landscapes, represented by random geometric graphs, by randomly drawing the locations of Z patches from a uniform distribution between 0 and 1 for x- and y-coordinates, respectively.DispersalWe model dispersal between local communities as a dynamic process of emigration and immigration, assuming dispersal to occur at the same timescale as the local population dynamics. Thus, biomass flows dynamically between local populations and the dispersal dynamics directly influence local population dynamics and vice versa. We model a hostile matrix between habitat patches that does not allow for feeding interactions to occur during dispersal. The total rate of emigration of animal species i from patch z is$${E}_{i,z}={d}_{i,z}{B}_{i,z}$$
(9)
with di,z as the corresponding per capita dispersal rate. We model di,z as$${d}_{i,z}=frac{a}{1+{{{{{{rm{e}}}}}}}^{-b({x}_{i}-{v}_{i,z})}}$$
(10)
with a, the maximum dispersal rate, b = 10, a parameter determining the shape of the dispersal rate, xi, the inflection point determined by the metabolic demands per unit biomass of species i, and υi,z, the net growth rate of species i on patch z. The net growth rate consists of the biomass gain by feeding, the biomass loss by being fed upon and the metabolic loss (({v}_{i,z}=frac{{B}_{i,z}mathop{sum}limits_{j}{e}_{j}{F}_{{ij},z}-mathop{sum}limits_{j}{{B}_{j,z}F}_{{ji},z}-{x}_{i}{B}_{i,z}}{{B}_{i,z}})). We chose to model di,z as a function of each species’ net growth rate to account for emigration triggers, such as resource availability, predation pressure, and inter- and intra-specific competition. If for example an animal species’ net growth is positive, there is no need for dispersal and emigration will be low. However, if the local environmental conditions deteriorate, the growing incentives to search for a better habitat increase the fraction of individuals emigrating.ImmigrationThe rate of immigration of biomass density of species i into patch z follows$${I}_{i,z}=mathop{sum}limits_{n,epsilon, {N}_{z}}{E}_{i,n}{max }(1-{delta }_{i,{nz}},0)frac{{max }(1-{delta }_{i,{nz}},0)}{mathop{sum}limits_{m,epsilon, {N}_{n}}{max }(1-{delta }_{i,{nz}},0)}$$
(11)
where Nz and Nn are the sets of all patches within the dispersal range of species i on patches z and n, respectively. In this equation, Ei,n is the emigration rate of species i from patch n, ({max }(1-{delta }_{i,{nz}},0)) is the fraction of successfully dispersing biomass, i.e. the fraction of biomass not lost to the matrix, and δi,nz is the distance between patches n and z relative to species i’s maximum dispersal distance δi (see below paragraph Maximum dispersal distance). The term (frac{{max }(1-{delta }_{i,{nz}},0)}{mathop{sum}limits_{m,epsilon, {N}_{n}}{max }(1-{delta }_{i,nz},0)})determines the fraction of biomass of species i emigrating from source patch n towards target patch z. This fraction depends on the relative distance between the patches, δi,nz, and the relative distances to all other potential target patches m of species i on the source patch n, δi,nm. Thus, the flow of biomass is greatest between patches with small distances to account for the logic that the first patch dispersing organism come across is closer. In other words, the further a destination is, the more likely it is to come across another patch before.For numerical reasons, we did not allow for dispersal flows with Ii,z  More

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