A population density, $u_{n}(x)$, is recursively defined by the formula egin{equation*} u_{n+1}(x)=int K(x-y)Big(1-ig(u_{n}(yig)Big)ig(u_{n} (yig)dy + ig(u_{n}(xig)ig(u_{n}(xig). end{equation*} Here, $K$ is a probability density function, $g(u)$ represents the fraction of the population that does not migrate, and $f$ is a monotonically decreasing function that behaves like the Beverton-Holt function. In this paper, I examine and modify the population genetics model found in cite{LV06} to include the case where a density-dependent fraction of the population does not migrate after the selection process.Using the expanded model, I developed a numerical application to simulate the spreading of a species and estimate the spreading speed of the population. The application is tested under various model conditions which include both density-dependent and density- independent dispersal rates. For the density-dependent case, I analyzed the fixed points of the model and their relationship to whether a given species will spread.

Large-scale data management systems rely more and more on cloud storage, where the need for efficient search capabilities clashes with the need for data confidentiality. Encryption and efficient accessibility are naturally at odds, as for instance strong encryption necessitates that ciphertexts reveal nothing about underlying data. Searchable encryption is an active field in cryptography studying encryption schemes that provide varying levels of efficiency, functionality, and security, and efficient searchable encryption focuses on schemes enabling sub-linear (in the size of the database) search time. I present the first cryptographic study of efficient searchable symmetric encryption schemes supporting two types of search queries, range queries and error-tolerant queries. The natural solution to accommodate efficient range queries on ciphertexts is to use order-preserving encryption (OPE). I propose a security definition for OPE schemes, construct the first OPE scheme with provable security, and further analyze security by characterizing one-wayness of the scheme. Efficient error-tolerant queries are enabled by efficient fuzzy-searchable encryption (EFSE). For EFSE, I introduce relevant primitives, an optimal security definition and a (somewhat space-inefficient, but in a sense efficient as possible) scheme achieving it, and more efficient schemes that achieve a weaker, but practical, security notion. In all cases, I introduce new appropriate security definitions, construct novel schemes, and prove those schemes secure under standard assumptions. The goal of this line of research is to provide constructions and provable security analysis that should help practitioners decide whether OPE or FSE provides a suitable efficiency-security-functionality tradeoff for a given application.