We test the controversial ideas about the role of corridors in fragmented animal habitats. Using simulation studies we analyze how fragmentation affects a simple prey-predator system and how the introduction of openings that connect the habitats changes the situation. Our individual based model consists of 3 levels: renewable prey food, as well as prey and predators that both have a simple economy. We find, in line with intuition, that the fragmentation of a habitat has a strong negative effect especially on the predator population. Connecting the fragmented habitats facilitates predator (and hence prey) survival, but also leads to an important counterintuitive effect: in the presence of a high quality predator, connected fragmented systems fare better in terms of coexistence than do unfragmented systems. Using a frequency domain analysis we explain how corridors between sub-habitats serve as "wave breakers" in the population flow, thus preventing deadly density waves to occur.

Spontaneous order happens in non-equilibrium systems composed of interacting elements. This phenomenon manifests in both the formation of space-time patterns in reaction-diffusion systems and collective rhythmic behaviors in coupled oscillators. In this thesis, we present the results of two studies: 1) The response of a multi-species predator-prey system to perturbation. 2) The features of a rich attractor space in a system of repulsively coupled Kuramoto oscillators. In the first part, we address this question: how does a complex coarsening system with non-trivial in-domain dynamics respond to perturbations? We choose a cyclic predator-prey model with six species each attacking three others. As a result of this interaction network, two competing domains form, while inside each domain three species play a rock-paper-scissors game which results in the formation of spirals inside the domains. We perturb the system by changing the interaction scheme which leads to a change of alliances and therefore a different spatial pattern. As expected, perturbing a complex space-time pattern results in a complex response. In the second part, we explore the attractor space of a system of repulsively coupled oscillators with non-homogeneous natural frequencies on a hexagonal lattice. Due to the negative coupling and the particular choice of geometry, some of the links between oscillators become frustrated. Coupled oscillators with frustration show similar features as frustrated magnetic systems. We use the parameters of the system like the coupling constant and the width of the frequency distribution to understand the system's attractor space. Further, we study the effects of external noise on the system. We also identify the breaking of time-translation invariance in the absence of external noise, in our system. / Doctor of Philosophy / Spontaneous ordering is a ubiquitous phenomenon observed in natural systems containing many interacting elements. In some systems the order is observed in the form of spatial patterns. It also can be seen in a population of coupled oscillators in the form of collective rhythmic behaviors. In this thesis, we present the results of two studies. For the first study, we choose a cyclic predator-prey system that shows a non-trivial space-time pattern. The system consists of six species each attacking three others, cyclically. By choosing such an interaction network, two competing domains form, while inside each domain three species play a rock-paper-scissors game. As a result of the inner competition, spirals form inside the domains. We study the response of the system to a perturbation. To perturb the system, we change the interaction scheme which leads to a change of alliances and therefore, a different spatial pattern. In the second study, we explore the patterns of clustering and synchronization in a system of repulsively coupled oscillators with non-homogeneous natural frequencies. Due to the negative coupling and the particular choice of geometry, some of the links between oscillators become frustrated. We use the parameters of the system such as the coupling constant and the width of the frequency distribution to understand the system’s attractor space. Further, we examine the effect of external noise on the system.

Human activity modifies the behaviour of large vertebrates and their acquisition of key resources. Despite the predation risk and competition for similar food resources that humans impose, wildlife consumers must acquire key foods across the landscape. Predation risk can modify foraging behaviour, yet we know little about the potential consequences, especially on large spatial scales. Humans may also affect food availability for wildlife by competing for shared prey, which most current harvest prescriptions fail to recognize. Against this background of threats to consumer-resource interactions, my research employed new conceptual, analytical, and practical approaches to seek not only new generalizable insight but also applied solutions. Addressing these goals, I characterized foraging behaviour by grizzly bears (Ursus arctos horribilis) on a focal prey, Pacific salmon (Oncorhynchus spp.), at multiple spatial scales. I predicted how human activity – both as modifications to landscapes and as salmon harvest – might affect bear-salmon interactions. I co-conceived, designed, and carried out this work through a framework of community engagement, which I crafted in collaboration with Indigenous communities in coastal British Columbia (BC). The framework (Chapter 2) identifies how scientists and communities can engage throughout the research process to work towards shared priorities, despite potential challenges in differences of knowledge systems or capacities. Methodologically, I used ratios of stable carbon (δ13C) and nitrogen (δ15N) isotopes in bear hair to estimate relative contributions of salmon in the annual diet of bears and employed existing data on landscape modification and salmon fisheries (i.e., escapement and catch) to characterize human activity and to measure associated variation in salmon consumption by bears. My first empirical contribution (Chapter 3) characterized spatial patterns of annual salmon consumption by grizzly bears across BC. I found substantial differences in salmon consumption within and among grizzly and black (U. americanus) bears in a large coastal region and across BC. Visualizing variation in consumer-resource interactions could guide conservation and management efforts that seek to protect predator-prey associations and marine subsidies for terrestrial ecosystems. I also investigated potential drivers of salmon consumption by bears in interior and coastal watersheds that varied in disturbance (Chapter 4). I found that human footprint in riparian areas of salmon-bearing watersheds affected bear diets more than the amount of salmon biomass available, showing that human activity can disrupt an otherwise strong predator-prey association. My community-based research occurred at the scale of a single large watershed, where I demonstrated how the Wuikinuxv First Nation might design their salmon management prescriptions according to their cultural values (Chapter 5). Despite a reduced abundance of salmon in the area, I identified harvest options that would trade-off benefits to local people and bears equally. In general, my dissertation research contributes to our understanding of the role humans increasingly play in mediating consumer-resource interactions. I also highlight how scientific research can support the leadership that local management can provide in mitigating human impacts to sustain an iconic predator-prey interaction of ecological, economic, and cultural importance. / Graduate / 2020-04-23

A spatiotemporal system of partial differential equations is implemented for describing a marine predator-prey system of shark and prey fish. The model is developed to account for predator migration and for harvesting of both predator and prey animals. The Finite Difference Method is employed to develop a numerical model to describe the behavior of the system in space over time. The dynamics of the system for different initial conditions for predator and prey populations and harvesting rates of both predators and prey using the numerical scheme. The resulting dynamics of the system from adding a predator sanctuary (an area within which the predator cannot be harvested) are also examined. It is hoped that this paper will illustrate that model behaves as a predator-prey system is expected to behave under the tested conditions. / acase@tulane.edu

We establish a theorem on the approximation of the solutions of delay differential equations with continuously distributed delay with solutions of delay differential equations with discrete delays. We present numerical simulations of the trajectories of discrete delay differential equations and the dependence of their behavior for various delay amounts. We further simulate continuously distributed delays by considering discrete approximation of the continuous distribution.

>Magister Scientiae - MSc / In principle, an enrichment of resources in predator-prey systems show prompts destabilisation of a framework, accordingly, falling trophic communication, a phenomenon known to as the \Paradox of Enrichment" [54]. After it was rst genius postured by Rosenzweig [48], various resulting examines, including recently those of Mougi-Nishimura [43] as well as that of Bohannan-Lenski [8], were completed on this problem over numerous decades. Nonetheless, there has been a universal none acceptance of the \paradox" word within an ecological eld due to diverse interpretations [51]. In this dissertation, some theoretical exploratory works are being surveyed in line with giving a concise outline proposed responses to the paradox. Consequently, a quantity of di usion-driven models in mathematical ecology are evaluated and analysed. Accordingly, piloting the way for the spatial structured pattern (we denote it by SSP) formation in nonlinear systems of partial di erential equations [36, 40]. The central point of attention is on enrichment consequences which results toward a paradoxical state. For this purpose, evaluating a number of compartmental models in ecology similar to those of [48] will be of great assistance. Such displays have greater in uence in pattern formations due to diversity in meta-population. Studying the outcomes of initiating an enrichment into [9] of Braverman's model, with a nutrient density (denoted by n) and bacteria compactness (denoted by b) respectively, suits the purpose. The main objective behind being able to transform [9]'s system (2.16) into a new model as a result of enrichment. Accordingly, n has a logistic- type growth with linear di usion, while b poses a Holling Type II and nonlinear di usion r2 nb2 [9, 40]. Five fundamental questions are imposed in order to address and guide the study in accordance with the following sequence: (a) What will be the outcomes of introducing enrichment into [9]'s model? (b) How will such a process in (i) be done in order to change the system (2.16)'s stability state [50]? (c) Whether the paradox does exist in a particular situation or not [51]? Lastly, (d) If an absurdity in (d) does exist, is it reversible [8, 16, 54]? Based on the problem statement above, the investigation will include various matlab simulations. Therefore, being able to give analysis on a local asymptotic stability state when a small perturbation has been introduced [40]. It is for this reason that a bifurcation relevance comes into e ect [58]. There are principal de nitions that are undertaken as the research evolves around them. A study of quantitative response is presented in predator-prey systems in order to establish its stability properties. Due to tradeo s, there is a great likelihood that the growth rate, attack abilities and defense capacities of species have to be examined in line with reviewing parameters which favor stability conditions. Accordingly, an investigation must also re ect chances that leads to extinction or coexistence [7]. Nature is much more complex than scienti c models and laboratories [39]. Therefore, di erent mechanisms have to be integrated in order to establish stability even when a system has been under enrichment [51]. As a result, SSP system is modeled by way of reaction-di usion di erential equations simulated both spatially and temporally. The outcomes of such a system will be best suitable for real-world life situations which control similar behaviors in the future. Comparable models are used in the main compilation phase of dissertation and truly re ected in the literature. The SSP model can be regarded as between (2018-2011), with a stability control study which is of an original.

This thesis focuses on the impact of spatially heterogeneous environments on the spatio-temporal behavior of planktonic systems. Specific emphasis placed is on the influence of spatial variations in the strength of random or chaotic movements (diffusion) of the organisms. Interaction between different species is described by ordinary differential equations. In order to describe movements in space, reaction–diffusion or advection–reaction–diffusion systems are studied. Examples are given for different approaches of diffusive motion as well as for the possible effects on the localized biological system. The results are discussed based on their biological and physical meanings. In doing so, different mechanisms are shown which are able to explain events of fast plankton growth near turbulent flows. In general, it is shown that local variation in the strength of vertical mixing can have global effects on the biological system, such as changing the stability of dynamical solutions and generating new spatiotemporal behavior. The thesis consists of five chapters. Three of them have been published in international peer-reviewed scientific journals. Chapter 1. Introduction: This chapter gives a general introduction to the history of plankton modeling and introduces basic ideas and concepts which are used in the following chapters. Chapter 2. Fokker-Planck law of diffusion: The influence of spatially in- homogeneous diffusion on several common ecological problems is analyzed. Dif- fusion is modeled with Fick’s law and the Fokker–Planck law of diffusion. A discussion is given about the differences between the two formalisms and when to use the one or the other. To do this, the discussion starts with a pure diffusion equation, then it turns to a reaction–diffusion system with one logistically growing component which invades the spatial domain. This chapter also provides a look at systems of two reacting components, namely a trimolecular oscillating chemical model system and an excitable predator–prey model. Contrary to Fickian diffusion, spatial inhomogeneities promote spatial and spatiotemporal pattern formation in the case of Fokker–Planck diffusion. A slightly modified version of this chapter has been published in the Journal of Mathematical Biology (Bengfort et al., 2016). Chapter 3. Plankton blooms and patchiness: Microscopic turbulent motions of water have been shown to influence the dynamics of microscopic species. Therefore, the number, stability, and excitability of stationary states in a predator– prey model of plankton species can change when the strength of turbulent motions varies. In a spatial system these microscopic turbulent motions are naturally of different strength and form a heterogeneous physical environment. Spatially neighboring plankton communities with different physical conditions can impact each other due to diffusive coupling. It is shown that local variations in the physical conditions can influence the global system in the form of propagating pulses of high population densities. For this, three local predator–prey models with different local responses to variation in the physical environment are considered. The degree of spatial heterogeneity can, depending on the model, promote or reduce the number of propagating pulses, which can be interpreted as patchy plankton distributions and recurrent blooms. This chapter has been published in the Journal Ecological Complexity (Bengfort et al., 2014). Chapter 4. Advection–reaction–diffusion model: Here, some of the models introduced in chapter 1 and 2 are modified to perform two dimensional spatial simulations including advection, reaction and diffusion. These models include assumptions about turbulent flows introduced in chapter 1. Chapter 5. Competition: Some plankton species, such as cyanobacteria, have an advantage in competition for light compared to other species because of their buoyancy. This advantage can be diminished by vertical mixing in the surround- ing water column. A non–spatial model, based on ordinary differential equations, which accounts for this effect is introduced. The main aim is to show that vertical mixing influences the outcome of competition between different species. Hystersis is possible for a certain range of parameters. Introducing a grazing predator, the system exhibits different dynamics depending on the strength of mixing. In a diffusively coupled horizontal spatial model, local vertical mixing can also have a global effect on the biological system, for instance, destabilization of a locally stable solution, or the generation of new spatiotemporal behavior. This chapter has been published in the Journal Ecological Modelling (Bengfort and Malchow, 2016).

In this PhD thesis, the utilisation of interactive simulation in a higher education e-learning classroom environment was explored and its effectiveness was experimentally evaluated by engaging university students in a classroom setting. Two case studies were carried out for the experimental evaluation of the proposed novel interactive simulation e-learning tool. In the first case study, the use of interactive agent-based simulation was demonstrated in teaching complex adaptive system concepts in the area of ecology to university students and its effectiveness was measured in a classroom environment. In a lab intervention using a novel interactive agent-based simulation (built in NetLogo). For the purpose of teaching complex adaptive systems such as the concept of spatially-explicit predator prey interaction to undergraduate and postgraduate students in the University of Stirling. The effectiveness of using the interactive simulation was investigated by using the NetLogo software and compared with non-interactive simulation built using R programming language. The experimental evaluation was carried out using a total of 38 students. Results of this case study demonstrates that the students found interactive agent-based simulation to be more engaging, effective and user friendly as compare to the non-interactive simulation. In the second case study, a novel interactive simulation game was developed (in NetLogo) and its effectiveness in teaching and learning of complex concepts in the field of marine ecology was demonstrated. This case study makes a twofold contribution. Firstly, the presentation of a novel interactive simulation game, developed specifically for use in undergraduate and postgraduate courses in the area of marine ecology. This novel interactive simulation game is designed to help learners to explore a mathematical model of fishery population growth and understand the principles for sustainable fisheries. Secondly, the comparison of two different methods of using the interactive simulation game within the classroom was investigated: learning from active exploration of the interactive simulation game compared with learning from an expert demonstration of the interactive simulation game. The case study demonstrated the effectiveness of learning from passive viewing of an expert demonstration of the interactive simulation game over learning from active exploration of the interactive simulation game without expert guidance, for teaching complex concepts sustainable fishery management. A mixed methods study design was used, using both quantitative and qualitative methods to compare the learning effectiveness of the two approaches, and the students’ preferences. The investigation was carried out by running interventions with a mixture of undergraduate and postgraduate students from the University of Stirling in a classroom environment. A total of 74 participants were recruited from undergraduate and postgraduate level for both case studies. This thesis demonstrated through two case studies effectiveness of the proposed novel interactive simulation in university e-learning classroom environment.