Threshold Resource Redistribution in Spatially-Structured Kinship Networks

Abstract

We present a model for a threshold-based resource redistribution process in a spatially-explicit population, characterizing the relation between kinship network structure, local interactions and persistence. We find that population survival becomes possible for lower resource densities, but leads to increased network heterogeneity and locally centralized clusters. We interpret this in relation to a feedback between the kinship network structure and reproduction ability. Agents receive stochastic resources and solicit additional resources from connected individuals when below a minimum, with each agent contributing a fraction of their excess based on relatedness. We first analyze a fully-connected population with uniform redistribution fraction and discuss mean field expectations as well as finite size corrections. We extend this model to a hub-and-spoke network, exploring the impact of network asymmetry or centrality on resource distribution. We then develop a spatially-limited population model with diffusion, local pairing, reproduction and mortality. Redistribution is introduced as a function of relatedness (generational distance through most-recent common ancestor) and distance. Redistribution-dependent populations exhibit a higher level of relational closeness with increased clustering for agents of highest node strength. These results highlight the interaction of resource density, cooperation and kinship in a spatially-limited regime.

Figures (8)

• Figure 1: Mean field survival fraction as a function of initial resource distribution mean and transfer fraction. We provide a simple demonstration of the expected mean field behavior for this threshold redistribution process. The surviving fraction is relative to the initial number of deficit agents. A threshold of one is assumed. If the resource mean is less than one, increased sharing is insufficient for full population survival. A larger resource mean requires a lower transfer fraction to effectively eliminate remaining downwards Poisson fluctuations.
• Figure 2: Survival fraction as a function of initial resource distribution mean and transfer fraction. We demonstrate the behavior of our threshold redistribution process for networks of 25 and 250 agents. A threshold of $\phi = 1$ is assumed. Results of these numerical simulations are plotted as a function of resource density for $\rho = 0.05, 0.1$ and 1.0 on the left. Additional, selected values are plotted on the right. With larger $N$, there are additional opportunities for resource transfer, leading to a higher survival fraction for the same $\rho$ relative to the low-$N$ case. The shaded regions represent the 68$\%$ central containment percentiles of the resulting survival fractions of $10^{4}$ simulations. The mean (true to the colorbar) is plotted with a solid line, while the median is also shown with a dashed line to indicate skewness.
• Figure 3: Population density with the number of living pairs and new children per time step. Here, population density represents the total number of agents relative to the number of existing grid spaces. The plots reflect a single simulation with instantaneous statistics from each time step after $\Delta T = 250$. Relative to agent mobility and the small map size, the population has reached an effective steady state after this period. High population densities indicate a large number of pairs and a small number of new children within the time step. This is a spatially-limited population. High density reduces population growth.
• Figure 4: Resource-dependent population density with the number of post-redistribution living pairs and new children per time step. Here, population density represents the total number of agents relative to the number of existing grid spaces. Each plot row reflects a single simulation with instantaneous statistics from each time step after $\Delta T = 250$. The resource mean of $\mu \approx 1.8$ was determined numerically as a critical point for the system, separating extinction from population persistence. Prior to this critical point, a lower, non-zero population density may exist for an extended period before extinction through fluctuations. Just beyond the critical point, this population is mortality limited, and a higher number of pairs leads to a higher number of children. At higher $\mu$ values, the population is largely impacted by spatial saturation, and a larger number of pairs leads to a lower number of children.
• Figure 5: Population size and agent redistribution opportunity for several values of $A$. The case of $A = 0$ represents a resource-dependent population without redistribution. On the top, we provide population size as a function of $\mu$ at late time, $\Delta T = 2000$. With each resource value and redistribution scheme, we plot the median and $98 \%$ central containment region of an ensemble of 50 simulations. The transition to a persistent population is rapid, and the course resource-mean sampling only serves to give a rough localization of $P_{\textrm{critical}}$. Once a persistent, stable population is possible, the median population size increases with $\mu$ with the transition towards spatial saturation. On the bottom, we provide several distributions of summed agent redistribution opportunity, $\sum_{j} \rho_{j}|_{A = 1}$. The labeled value of $A$ within the legend represents the value used in simulation, while $A = 1$ is evaluated here for clearer statistical comparison. The resulting distributions reflect only agent generational and spatial familial proximity. These are aggregate distributions from ensembles of 200 simulations. We find that these distributions are lifted towards higher values when redistribution is possible, suggesting more organized or clustered populations.