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Ecosystem conservation is fundamental to guarantee the survival of endangered species and to preserve other ecological functions important for human systems (e.g., water). Planning land conservation increasingly requires a landscape approach to mitigate the negative impacts of spatial threats such as urbanization, agricultural development, and climate change. In this context, landscape connectivity and compactness are vital characteristics for the effective functionality of conservation areas. Connectivity allows species to travel across landscapes, facilitating the flow of genes across populations from different protected areas. Compactness measures the spatial dispersion of protected sites, which can be used to mitigate risk factors associated with species leaving and reentering the reserve. This research describes an optimization model for the design of conservation areas, while inducing connectivity and compactness. We use the Reocks index, a metric of compactness that maximizes the ratio of area of the selected patches to the area of their smallest circumscribing circle. Our model includes budget and minimum protected area constraints to reflect realistic financial and ecological requirements. The initial nonlinear model is reformulated into a mixed-integer linear program, which is solved using an adaptation of the Newtons method for problems with integer variables. We characterize an optimal solution and derive cuts to improve the model performance. We illustrate our results using real life landscapes with irregular patches. 

Abstract Genetic connectivity lies at the heart of evolutionary theory, and landscape genetics has rapidly advanced to understand how gene flow can be impacted by the environment. Isolation by landscape resistance, often inferred through the use of circuit theory, is increasingly identified as being critical for predicting genetic connectivity across complex landscapes. Yet landscape impediments to migration can arise from fundamentally different processes, such as landscape gradients causing directional migration and mortality during migration, which can be challenging to address. Spatial absorbing Markov chains (SAMC) have been introduced to understand and predict these (and other) processes affecting connectivity in ecological settings, but the relationship of this framework to landscape genetics remains unclear. Here, we relate the SAMC to population genetics theory, provide simulations to interpret the extent to which the SAMC can predict genetic metrics and demonstrate how the SAMC can be applied to genomic data using an example with an endangered species, the Panama City crayfish Procambarus econfinae , where directional migration is hypothesized to occur. The use of the SAMC for landscape genetics can be justified based on similar grounds to using circuit theory, as we show how circuit theory is a special case of this framework. The SAMC can extend circuit‐theoretic connectivity modelling by quantifying both directional resistance to migration and acknowledging the difference between migration mortality and resistance to migration. Our empirical example highlights that the SAMC better predicts population structure than circuit theory and least‐cost analysis by acknowledging asymmetric environmental gradients (i.e. slope) and migration mortality in this species. These results provide a foundation for applying the SAMC to landscape genetics. This framework extends isolation‐by‐resistance modelling to account for some common processes that can impact gene flow, which can improve predicting genetic connectivity across complex landscapes.