The dynamics and interactions among species in the ecosystem with their complex behaviour are the principal part of mathematical ecology and are the concern of many biological and ecological processes. An ecosystem is the biological community of all species of a given area and their physical environment [1] [2]. The application of mathematical theories and concepts in ecology has resulted in another branch of biology known as mathematical ecology. This studies the time dependent behaviour of modelled ecological systems [3] [4] [5]. These models give us significant insights for the behaviour of the system and nature. Usually these systems consider various complex communities involving various species which interact in a very complicated way and hence, challenging to analyse and draw conclusion about them. Mathematical models can govern the time evolution of species in terms of their size, age, interactions, food supply, disease and environmental conditions including seasonal and climatic variations. Since all organisms in nature are interdependent [6], the interaction existing among them influences directly and indirectly the survival of each organism and the performance of the ecosystem in general. Hence, studying the comprehensive behaviour and dynamics of the ecosystem is vital for maintaining scientific management, ecological integrity of species and conservation of the ecosystem [7] [8] [9]. There are various ways in which living things can interact, that is by commensalism, competition, predation and mutualism [10]. However, the interaction between predator and their own prey (prey-predator interactions) continues to be one of the dominant areas of research in mathematical ecology due to its universal existence, complexity and importance [11] [12]. For this reason, we believe that, this review may be of particular interest to those researchers involved in modelling this kind of biological and ecological problems allowing them to have an overview of the work which is currently being carried out. Therefore, our objective in this paper is to analyse several adaptations of prey-predator models that have been done in recent years to model different real situations that appear in the field of population dynamics.

It is known that, predator population always depends on their prey species as a source of food, hence for survival [13] [14]. The interaction of converting prey biomass into predator biomass through predation has the direct effect between predators and prey and has been a focus of mathematical modelling on prey-predator interactions [15] [16]. Interaction between prey-predator is complex process and non-linear process and hence understanding various biological phenomena in the dynamics of predator-prey systems is vital in ecology and conservation biology. Just under a century ago a very first endeavour to predict the dynamics of species in the ecosystem was predicted mathematically. The physician [17] and mathematician [18] were the first to develop a simple model explaining population dynamics in the context of competing species. At first Lotka developed a simple model on animal and plant species and then he developed the application of the study of the dynamics of a prey-predator system [19]. On his behalf Volterra considered the same model simultaneously to explain the observation about the percentage of predatory fish caught during the years of World War I which had increased. This fact seemed confusing because the fishing effort had been reduced during those years. So Volterra was interested in studying that situation. To explain this behaviour, Volterra stated a simple system of ordinary differential equations.

d N d t = N ( a − b P ) d P d t = P ( − c + d N ) (1)

where N and P are the densities of prey of prey and predator respectively, a is the growth rate of prey, c is the mortality rate of predator where as b is the predation rate and d is the conversion rate. Later Lotka-Volterra systems were generalized and considered in arbitrary dimension.

d U i d t = U i ( b i 0 + ∑ j = 1 n     b i j U j ) ,   i = 1 , ⋯ , n (2)

Consequently, the applications of this system started to multiply. New systems on prey-predator dynamics had been formulated but also these systems has been used for modelling many other natural phenomena such as chemical reaction [20], plasma physics [21], hydrodynamics [22] and other problems from economics and social science [23]. In a view of this, it is clear that population models in particular competition or prey-predator model are important both in their original field and in their application to many other problems from other areas. Their study is also interesting from theoretical point of view. Actually the study of prey-predator models has always been one of the hot sports in biomathematics, so there is a very big number of works on this topic, especially in dimension two. Table 1 shows the list of some reviewed prey-predator models that considered the aspects of seasonal weather variation and disease infections.

In literature, one of the first contributions to the study of prey-predator systems was in [8] who considered the dynamics of lion, wildebeest and zebra in Kruger national park, lion being a predator while wildebeest and zebra are prey species.

d x 1 d t = x 1 ( b 1 − a 12 x 2 + a 13 x 3 ) d x 2 d t = x 2 ( − b 2 − a 21 x 1 + a 23 x 3 ) d x 3 d t = x 3 ( b 3 − a 32 x 2 + a 33 x 3 ) (3)

where x₁, x₂ and x₃ represents the population density of wildebeest, zebra and lion respectively. Stability of the equilibrium point was analysed by eigenvalue approach. However we see that the author did not consider factors that can affect

species such as poaching, seasonal weather variations, disease and fire. Also the study considered the linear function response ignoring the logistic growth term. The same style of modelling was also seen in [38] who developed the three species prey-predator systems as follows,

d x 1 d t = x 1 [ λ 1 − α 12 x 2 + α 13 x 3 ] − q 1 E x 1 d x 2 d t = x 2 [ λ 2 − α 12 x 2 + α 13 x 3 ] − q 1 E x 1 d x 3 d t = x 3 [ λ 3 − α 12 x 2 + α 13 x 3 ] − q 1 E x 1 (4)

where x₁, x₂ and x₃ represents the population density of prey species one, prey species two and predator respectively. This model differ from the previous by harvesting term q i E x 1 . The author considered harvesting as an activity that can stabilize or distabilize the ecosystem.

The two models above assume that prey species can grow infinitely but also the level of consumption of prey species by predator is exponential. In many cases, this may not represent the reality in many ecosystems. In order to avoid unreliable results, more general and realistic models have been suggested by considering the logistic growth to prey species and functional responses to predator (predator’s instantaneous per capita feeding rate as a function of prey abundance). Example, [9] formulated and analysed the mathematical model for the study of the interaction of lion, buffalo and Uganda kobs of the Queen Elizaberth national park in Uganda.

d N 1 d t = r 1 N 1 ( 1 − N 1 K 1 ) − α 1 N 1 N 2 − ( a 1 N 1 1 + b 1 N 1 ) P − E N 1 d N 2 d t = r 2 N 2 ( 1 − N 2 K 2 ) − α 2 N 1 N 2 − ( c N 2 1 + d 1 N 2 + d 2 P ) P d N 3 d t = − e P − ( a 1 N 1 1 + b 1 N 1 ) P + ( c N 2 1 + d 1 N 2 + d 2 P ) P (5)

Here the author assumed that prey species are growing logistically by including the constant carrying capacity, also used two kinds of function responses, the hyperbolic function response on the first prey and ratio-dependent function response on the second prey. Both local and global stability of equilibrium points was analysed. Once of the important results they obtained is that, all population would co-exist if any removal (death rate, harvesting and predation) were not below the intrinsic growth rate of every species. The same results were also obtained in [12] where was modelling the interaction of cichlid fishes, tilapia fish and Nile perch of lake Victoria ecosystem where they formulated the following model.

d x 1 d t = λ 1 x 1 ( 1 − x 1 K 1 ) − α 12 x 1 x 2 − α 13 x 1 x 3 1 + β x 1 − q 1 E x 1 d x 2 d t = λ 2 x 2 ( 1 − x 2 K 2 ) − α 21 x 1 x 2 − α 23 x 2 x 3 1 + γ x 2 − q 2 E x 2 d x 3 d t = − w x 3 + α 31 α 13 x 1 x 2 1 + β x 1 − α 32 α 23 x 2 x 3 1 + γ x 2 − q 3 E x 3 (6)

As in [9], here the author considered the logistic growth to prey species and the Holling type II function response on predator. Here the author also assumed that every was harvested. Both [9] and [12] tried to model ecosystems by including more important factors. Other reviewed literature of the same scheme of modelling as the previous are [38] [39] [40]. Some considered prey refuge while others considered the reserved zones, time delay and alternative food.

According to [41] [42], Serengeti ecosystem which is in latitude 10S to 3030'S, and longitude 340E to 360E in east Africa, spanning northern Tanzania and some parts of southwestern Kenya was officially found in 1951 and is now the Tanzanians largest national park. It is extraordinary in species and it is one of the most popular wildlife reserves in the world. It has a lot of terrestrial mammal migration which makes it to be among the seven wonders of Africa [41] [42] [43]. This high diversity of the ecosystem is due to diverse habitats, including forests, kopjes, woodlands, swamps and grassland. The ecosystem comprise also of extraordinary animal species such as spotted hyenas, elephants, giraffe, thomsons gazelles, zebra, wildebeest, leopards, lion, wild dogs, impala, buffalo, Coke’s hartebeest, cheetah, grant’s gazelle, topi and more than five hundred species of birds [19] [41]. Despite of the vital modelling work achieved by various researchers, there has been far less quantitative focused work on predation and competition of biological species in Serengeti ecosystem particularly the interaction of wildebeest, zebra and lion which are keystone species of the Serengeti ecosystem. However, few scholars studied about the prey-predator interaction in the ecosystem such as [44] [45] [46] [47]. Example, Sagamiko et al. (2015) developed the following model to study the threats to Lionwildebeest Predator-Prey dynamics with Optimal control in Serengeti national park.

d x d t = r x ( 1 − x k ) − w x y x + a − p x − ( 1 − f ) x d y d t = − a 2 y + w 1 x y x + a − c y − ( 1 − e ) y (7)

It is a two species predator-prey system where predator was lion and prey was wildebeest. They also used the optimal control theories to establish the optimal strategies for controlling the threats in the system where for poaching, the strategy was ant-poaching patrols, for retaliatory killing was construction of strong bomas and for drought they proposed then construction of dams. They found that the best result is achieved when all three controls are simultaneously used and the combined is a good approach for the scientific management of the ecosystem.

Another researcher to model prey-predator systems in Serengeti ecosystem is [19] who analysed the population dynamics of herbivores, carnivores and grasses volume using secondary data from the years 1996 to 2006. They found that, the crocodile predation on herbivore decreases the population of herbivore and the crocodile increases when the natural death rate in the absence of prey decreases. Also they found that herbivore population increase as its intrinsic logistic rate increases.

[41] also developed a comprehensive model to study the dynamics of the Serengeti-Mara ecosystem. The processes of the research involved first studying the spatial and temporal variation in climatic parameters and physiography around the Serengeti ecosystem. The relationship between rainfall and grass growth was then involved and finally, the impact of predation to those prey species. The system, even in these early stages of formulation, adequately depicted dynamics that are equivalent to those in the Serengeti ecosystem, which shows that appropriate methods were used. It was also found that the grass availability was the main factor determining the overall dynamics of wildebeest in the entire system.

[46] studied the predation and competition of Serengeti migrants captured by hierarchical models. They statistically studied wildebeest and zebra migrating seasonally around the Serengeti-Mara ecosystem. They studied how food availability and predation determine the migration patterns of individual species in the ecosystem. They found that wildebeest tend to move with regard to food quality paying little attention to predation risks while the movement and zebra reflected a balance between the predation risk and the food quality food for sufficient biomass. Their analysis shows the way two migratory species migrates in response to different aspects in the same landscape.

More ever, [47] explains the presence of poaching in Serengeti ecosystem and protected areas which is accelerated by revenue generated from from bush meat sales. He compared the benefits and costs through monetary benefit, bodily injuries, fires and prison sentences that individual over goes over their poaching career.

Prey-predator models are widely studied topics in biomathematics. We have reviewed recent works focusing on the modelling of harvesting, seasonal weather variation and disease. Some of the stated models show a very rich dynamics which is much more realistic that the one of the first historical models. The inclusion of seasonal weather variation in prey-predator systems can have important effects on the dynamics. It can stabilize or distabilize the systems as seen in [25] and [48]. The conclusion about the influence of disease is not the same in all the studied models. Considering the system with infections in [28] the author shows that the population of prey species increases after the application of treatment. In [26] it is shown that the increase decreases the population and can even cause extinction. We noticed that the same methods and results are used in most papers for the study of the stability of the equilibria and the qualitative behaviour of the system. Principally the authors use the standard linear stability analysis, they analyze the sign of the real part of the eigenvalues of the Jacobian matrix, sometimes using Routh Hurwitz criterion. It is important to remember that for non-hyperbolic equilibria, the computation of the eigenvalues does not allow us to conclude on the stability. We have observed that, in such cases, the equilibria were not usually studied in the papers reviewed. Other frequently used tools are the computation of Lyapunov coefficients.

In most works, there are some problems, open problems and some characteristics that would be interesting to investigate more in detail. In the following we list some problems which researchers may find interesting to address in the future, which are proposed or motivated by the works included in this review.

· Among the systems that consider the harvesting, the one proposed in [9] is the one that shows a richer dynamics. As the authors say, it is possible that by changing the functional response and taking one of ratio dependent function response, the behaviour is even more realistic, so it would be interesting to study it.

· Many systems study infections on only prey population or predator population but it would be interesting to consider models with infections both in prey and predator populations, because as stated in [57], in many ecological systems, predator takes high parasites of infected prey. Among the systems that consider infections, the model proposed in [26] shows rich dynamics as it explores infections both on prey and predator as well as possibility of treatment on both populations.

· The effect of seasonal weather variation on prey-predator systems has not been studied as far as we know. Hence there is a need to study new functions that can model the effects of seasonal weather variations on prey-predator ecosystems, (see [25] and [29]).

We acknowledge the Dar es salaam Institute of Technology for their support.

The authors declare no conflicts of interest regarding the publication of this paper.

Charles, R., Makinde, O.D. and Kung’aro, M. (2022) A Review of the Mathematical Models for the Impact of Seasonal Weather Variation and Infections on Prey Predator Interactions in Serengeti Ecosystem. Open Journal of Ecology, 12, 718-732. https://doi.org/10.4236/oje.2022.1211041