New insight into the nano-fluid flow in a channel with tempered fractional operators.

While studying time fractional fluid flow problems it is typical to consider the Caputo derivative, however, these models have limitations including a singular kernel and an infinite waiting time from a random walk perspective. To help remedy this problem, this paper considers a tempered Caputo derivative, giving the system a finite waiting time. Initially, a fast approximation to a generalised tempered diffusion problem is developed using a sum of exponential approximation. The scheme is then proven to be unconditionally stable and convergent. The convergence properties are also tested on a sample solution. The fast scheme is then applied to a system of coupled tempered equations which describes the concentration, temperature and velocity of a nanofluid under the Boussinesq approximation. The most notable finding is that increasing both the fractional and tempering parameters reduces the heat transfer ability of the nanofluid system.

Choosing optimal trigger points for ex situ, in toto conservation of single population threatened species.

Many endangered species exist in only a single population, and almost all species that go extinct will do so from their last remaining population. Understanding how to best conserve these single population threatened species (SPTS) is therefore a distinct and important task for threatened species conservation science. As a last resort, managers of SPTS may consider taking the entire population into captivity-ex situ, in toto conservation. In the past, this choice has been taken to the great benefit of the SPTS, but it has also lead to catastrophe. Here, we develop a decision-support tool for planning when to trigger this difficult action. Our method considers the uncertain and ongoing decline of the SPTS, the possibility that drastic ex situ action will fail, and the opportunities offered by delaying the decision. Specifically, these benefits are additional time for ongoing in situ actions to succeed, and opportunities for the managers to learn about the system. To illustrate its utility, we apply the decision tool to four retrospective case-studies of declining SPTS. As well as offering support to this particular decision, our tool illustrates why trigger points for difficult conservation decisions should be formulated in advance, but must also be adaptive. A trigger-point for the ex situ, in toto conservation of a SPTS, for example, will not take the form of a simple threshold abundance.