Surface Temperature Influences the Population of Limnothrissa miodon in Lake Kariba

Global warming is a serious world problem where earth’s temperature has been reported to increase over the years; the aquatic ecosystems are also not the exceptions. But, the efects of this phenomenon on the aquatic ecosystems are not well understood. Tis study aims to understand the infuence of surface temperature on the population density of Limnothrissa miodon in Lake Kariba. We constructed a mathematical model on the population dynamics of Limnothrissa miodon with nutrients, phyto-plankton, zooplankton


Introduction
Global warming is one of the major issues confronting the modern civilization [1]. Te major concern is the increasing temperature of the biosphere; the aquatic ecosystems are not an exception. Major concern of the aquatic biologists is to understand the adverse efects of this phenomenon on the aquatic communities [2]. With this purpose, this study was conducted to determine the infuence of lake surface water temperature on the population density of Limnothrissa miodon in Lake Kariba. Te Lake Kariba fshery which is shared by both Zimbabwe and Zambia is of paramount importance for the continued survival of the small pelagic clupeid, Limnothrissa miodon [3]. Limnothrissa miodon (Boulenger, 1906), also referred to as kapenta, is a major source of protein and income for fshing cooperatives, wholesalers, retailers, and the local community [4]. Reports that the kapenta catches are in decline are a cause of concern to many stakeholders in the kapenta fshery and academia. Te decline in the catches in kapenta has been attributed to overfshing, predation by Hydrocynus vittatus, climate change, and the reduction of nutrient infow into Lake Kariba. According to Nhambura [5], several cooperatives and companies which make up the Kariba kapenta industry are struggling and facing collapse since the amounts of kapenta in Lake Kariba are dwindling because of the changing climate. Te depletion of kapenta may also have a negative efect on the levels of Hydrocynus vittatus (tigerfsh), which feeds on kapenta and contributes immensely to the economy of Kariba. Terefore, it is important that we formulate and analyse a mathematical model in order to understand the efects of temperature on the population density of kapenta in Lake Kariba.
Ndebele-Murisa [6], recorded the relative cellular concentration of phytoplankton classes in the Sanyati basin of Lake Kariba. Te highest concentration was in the less palatable Cyanophyceae (78.05%), followed by Chlorophyceae (9.7%), Dinophyceae (7.2%), Bacillariophyceae (1.7%), Euglenophyceae (1.6%), Chrysophyceae (1.3%), and Xanthophyceae (0.45%). Te species with the highest relative concentration of 66.20% was Cylindrospermopsis raciborskii, followed by Microcystis aeruginosa(2.79%) from the Cyanophyceae class. From the Chlorophyceae class, Staurastrum johnsonii(0.67%) had the highest relative concentration, followed by Coelastrummicroponum(0.61%) and Oedogonium sp. (0.60%). Ndebele-Murisa [6] also recorded the relative concentrations of the Chlorophyceae species Staurastrum johnsonii(0.67%), Coelastrum microponum(0.61%), and Oedogonium sp. (0.60%) in the Sanyati basin. Cynaphytes are associated with the production of toxins [7,8] and are often unigestible, indigestible, or nutritionally poor [9,10]. Many species of Chlorophytes are less palatable due to long splines and long processes [11]. In Lake Kariba, Limnothrissa miodon mainly feeds on zooplankton, especially the Cladoceran species Bosmina longirostris [12,13]. Machena [14] found out that 99% of the diet composition of Limnothrissa miodon is zooplankton. According to Mhlanga [15], Limnothrissa miodon is a major prey species of Hydrocynus vittatus. Kenmuir [16] found out that 70% of the diet of Hydrocynus vittatus consisted of Limnothrissa miodon. Environmental factors play an important role in a fshery [17][18][19], and as a result, it is critical to investigate mathematically the role of temperature in the dynamics of Limnothrissa miodon in the Lake Kariba fshery. Chifamba [17] found that maximum temperature was the best predictor of catch per unit efort (CPUE) of kapenta for the period 1970−1996 and suggested that kapenta catch cycles might be related to weather conditions. Magadza [18] estimated a 100−year warming of 4.8°C for the Lake Kariba area, using data from the Kariba meteorological station for the period 1965−2000. Te lake warmed by a mean of 1.54°C between 1965 and 1990, corresponding to a warming rate of 0.62°C per decade [18]. Magadza [18] used temperature data from 1965−2000 and determined an air temperature and water temperature maximum turning point of 34.8°C and 28.7°C, respectively, around 1987 − 88 for Limnothrissa miodon in Lake Kariba. Te kapenta catches increased exponentially prior to the turning point and then decreased linearly afterward. Regression analysis results from a study by Ndebele-Murisa et al. [19] showed that temperature is a major factor in kapenta catches from 1968−2008. Ndebele-Murisa et al. [20] found that maximum temperature contributed to 72% of the variation in kapenta stocks, 68% of kapenta CPUE, and 99% of the change in lake water levels from 1974 to 2010 in Lake Kariba. Sibanda [21] compared the temperature responses of selected phytoplankton classes in laboratory cultures, focusing on Chlorophyceae and Cyanophyceae. According to her fndings, the growth rate of Chlorophyceae decreased above 25°C, while that of Cyanophyceae, on the other hand, increased almost exponentially up to 34°C. Magadza [18] observed a transition from Chlorophyceae to Cyanophyceae in Lake Kariba and found out that Cyanophyceae, particularly Cylindrospermum raciborskii, now dominate the lake phytoplankton.
How the temperature describes and infuences the dynamics of the kapenta population in Lake Kariba has not been studied. A deterministic model involving nutrients, phytoplankton, zooplankton, kapenta, and tigerfsh, as well as temperature as an environmental factor, is yet to be developed and tested. Te kapenta model will aid in our understanding of the aquatic ecosystem dynamics in the kapenta fshery in Lake Kariba and we will be able to explain the infuence of lake surface water temperature on kapenta levels. We hypothesized that (i) a shift from Chlorophyceae to Cyanophyceae will lead to a decline in kapenta population density and (ii) warming of the lake has an adverse efect on the population density of kapenta. Te objectives of the study were to (i) formulate and analyse a mathematical model with data ftting, (ii) determine the optimum temperature for phytoplankton growth for the ftted lake surface water temperature data, (iii) compare numerical results on kapenta population density for the mathematical models with the Chlorophyceae and Cyanophyceae phytoplankton classes, and (iv) determine the efect of lake warming on kapenta population density.

Study
Area. Lake Kariba is located in a tropical area with seasonal rainfall on the Zambezi river between latitudes 16°28 ′ to 18°04 ′ S and longitudes 26°42 ′ to 29°03 ′ E [22], has a volume and surface area of 160 km 3 and 5364 km 2 , respectively, and has an average width of 19.4 km, with the widest section measuring 40 km [23]. Te lake is 486 m above sea level, and the shoreline is about 2164 km long [3,23]. Te kapenta fshery in Lake Kariba is highly mechanised, licence-controlled, and is shared by Zimbabwe and Zambia [6]. A map of Lake Kariba and its fshing basins is shown in Figure 1

Model Formulation
Te model has 5 classes whose densities are functions of time: N(t) denoting the concentration of nutrients, P(t) is the population density of phytoplankton, Z(t) is the zooplankton population density, L(t) is the density of the Limnothrissa miodon population, and R(t) is the density of the Hydrocynus vittatus population. Te Limnothrissa miodon model [26] is developed to include the efect of temperature on growth of phytoplankton. Te coefcient of temperature for the growth of phytoplankton [27] is assumed to be where T � T(t) is lake surface water temperature, T opt is optimal temperature for phytoplankton growth, and T min is the minimum temperature for phytoplankton growth. Te water temperature is modelled by where the parameters b, c, ω, and ϕ are estimated from data ftting. T(t) is the ftted temperature in C, b is the mean water temperature, c is the amplitude, ω is the angular frequency (2π/12), t is time in months, and ϕ is a phase angle. Te nonlinear dynamical system is with initial condition and defned as To be the mathematically feasible region, nutrients enter the water body at the rate π 0 where π 0 > 0 is a constant; β 1 is an uptake rate; and β 2 , β 3 , and β 4 are the predation rate International Journal of Ecology coefcients. Te α i 's for i � 0, 1, 2, 3, 4 are depletion rate coefcients. Te π i 's for i � 1, 2, 3, 4 are the phytoplankton, zooplankton, Limnothrissa miodon, and Hydrocynus vittatus growth rate coefcients, respectively. Te coefcient σ is a positive constant for the crowding of the Limnothrissa miodon population. Kapenta are harvested at a rate qEL, where q is the catchability coefcient and E is the efort measured as boat nights. Te growth rate of Hydrocynus vittatus is (π 4 ZL/d + L) [28] and d is the population density of kapenta at which specifc growth rate becomes half of its saturation value. Te tigerfsh are harvested at a rate κηR, where κ is the catchability coefcient and η is the efort.

Positivity of Solutions
and we obtain From the second equation of system (3), By integrating (8), we obtain From the third equation of model (3), and integrating (10) results in Considering the fourth equation of system (3), and integration of (12) results in According to the ffth equation of model (3), Integration of (14) yields We therefore conclude that solutions of model (3) with initial conditions (4) remain positive ∀t ≥ 0.

Data Fitting Results
A cosine function in equation (2) was ftted in MATLAB R2016a to the mean monthly surface water temperature data for the period July 2014 to December 2017 in order to estimate and describe the variation in the lake surface water temperature in Lake Kariba. Te output obtained from MATLAB was T(t) � 26.89 + 3.027 cos(−0.6267t − 11.49).
Te estimates of parameters of equation (2) are shown in Table 1. Te goodness of ft statistics for the ftted model (21) were sum of squared estimate of errors (SSE)� 1.423, R 2 � 0.97666, R 2 adj � 0.9678, and root mean square error (RMSE)� 0.4218.
Te actual and ftted surface water temperature plots are shown in Figure 2.
Te ftted model has a very high R 2 adj indicating that the ft quality to the series of the temperature data points is good.

Numerical Simulations
For the coefcient of temperature for the growth of phytoplankton, g(T), in (1) we assumed T min � 5°C [27], T opt � 25°C for Chlorophyceae, and T opt � 34°C for Cyanophyceae in Lake Kariba [18]. Te plots of the temperature coefcient g(T) versus temperature for Chlorophyceae and Cyanophyceae are shown in Figures 3(a) and 3(b), respectively.

Chlorophyceae and Cyanophyceae.
In this subsection, we compared numerical results on kapenta population density for the mathematical models with the Chlorophyceae and Cyanophyceae phytoplankton classes. We simulated model (3) using the ftted equation for T in (21) and the default parameter values in (22) to illustrate the efect of T opt on the dynamics of model (3).
Te parameter values in equation (22) were obtained from published data, while the others were estimates. For the numerical simulations, a fourth-order Runge-Kutta numerical scheme written in Wolfram Mathematica 11 was used. Te units of the variables N, P, Z, L, and R in the model system (3) are μgl − 1 . Numerical simulation results in Figures 4-6 show a decline in the population density of phytoplankton, zooplankton, kapenta, and tigerfsh as T opt changes from 25°C to 34°C.
Te time series plots and phase portraits in Figures 4-6 show oscillatory behaviour and a periodic orbit of period 1, respectively. Analysis results from the model (3) with the ftted surface water temperature (21) show that if T opt shifts from 25°C to 34°C, which corresponds to the shift in dominance of Cyanophyceae to Chlorophyceae, then there will be a decline in the population density of kapenta. So, if the less palatable Cyanophyceae species become dominant in Lake Kariba, we expect a decrease in kapenta abundance in the lake. Pliński and Jóźwiak [29] found that phytoplankton progressed from Bacillariophycea to Chlorophyceae and then Cyanophyceae as water temperature rose. According to Solomon et al. [30], the decline in zooplankton production was due to a decrease in the more palatable Chlorophyceae as a result of rising water temperatures. Ndebele-Murisa [6] recorded a relative cellular concentration of 78.05% and 9.7% for Cyanophyceae and Chlorophyceae, respectively, in Lake Kariba, showing the dominance of Cyanophyceae over Chlorophyceae. Simulation results in Figures 4-6 are therefore in agreement with fndings from Solomon et al. [30] and Magadza [18].

Temperature Changes in Lake Kariba.
In order to determine the optimum temperature for phytoplankton growth for the ftted lake surface water temperature data which result in the maximum kapenta population density, T opt in (2) varied from 25°C to 35°C using the ftted equation for T in (21) for model (3) with T min � 5°C. Te plot in magenta in Figure 7 shows the maximum and the curve in blue shows the minimum values of L(t).    International Journal of Ecology Figure 7 shows that the kapenta population density maximum curve had a maximum at about 30°C and declined thereafter. Terefore, based on that result, it can be concluded that the optimal temperature for kapenta is approximately 30°C. Te result is not far of from the breakpoint of 28.7°C for the kapenta catch obtained by Magadza [18].

International Journal of Ecology
In order to analyse model (3) for the warming of the lake water, we assumed that the lake warms by a rate of 0.62°C per decade [18]. We added 0.62°C to the mean surface water temperature data for the period 2014 − 2017 to obtain an estimate T 1 (t) of the mean surface water temperature after a decade. We projected further for the second decade by adding 0.62°C to T 1 (t) to get T 2 (t). We modelled the water temperatures T 1 (t) and T 2 (t) with (2), and the ftted models in (23) and (24) were T 2 (t) � 28.13 − 3.027 cos(−0.6267t + 4.218).   Te ftted models were then substituted into (1). We simulated model (3) using some initial condition and compared numerical results from model (3) using the 2014 − 17 data, T 1 (t), and T 2 (t), with T opt � 25°C and T min � 5°C in (1). Numerical results which illustrate the infuence of lake warming on the concentration of nutrients and population density of phytoplankton, zooplankton, kapenta, and tigerfsh are shown in Figures 8-10.
Results from Figures 8-10 show that as the lake warms, there is a corresponding decline in the population density of zooplankton, therefore resulting in a decline in the kapenta and tigerfsh population density in the lake. Terefore, warming of the lake may have a negative efect on the population density of kapenta in the Lake Kariba fshery, and the environmental factor temperature may have a signifcant impact on kapenta abundance.

Discussion
In light of global warming, there is a knowledge gap regarding the efects of temperature increase on the density of Limnothrissa miodon in Lake Kariba. In this paper, we formulated and analysed a model that has nutrients, phytoplankton, zooplankton and Limnothrissa miodon, and temperature as an environmental factor. Te phytoplankton growth rate, phytoplankton mortality, grazing on phytoplankton, zooplankton growth rate, zooplankton mortality, grazing on zooplankton, and Limnothrissa miodon mortality were assumed to be Holling type I forms. Feeding on Limnothrissa miodon and Hydrocynus vittatus growth was assumed to be Holling type II forms. Positivity and existence of solutions to model (3) were investigated. Numerical simulations were done for the model with default parameter values. Mathematical modelling provided an overall picture of the dynamics of the model variables. Assuming that existing trends in surface lake water temperature will continue, we used numerical simulations to demonstrate that global warming infuences the Limnothrissa miodon population density in Lake Kariba. Te thermal response of Limnothrissa miodon to global warming was considered in the context of thermal optima.
For the model with ftted lake surface water temperature data for the period 2014 − 2017, we used numerical simulations and investigated the efect of a shift in the optimal temperature for phytoplankton growth from 25°C to 34°, corresponding to dominance of Cyanophyceae over Chlorophyceae, and results showed a decline in the population density of Limnothrissa miodon.
Te optimum temperature for phytoplankton growth for the ftted lake surface water temperature data for the period 2014 − 2017 varied from 25°C to 35°C, and simulation results showed a maximum kapenta population density at approximately 30°C and the result is similar to the breakpoint of 28.7°C for kapenta catch obtained by Magadza [18]. According to Abowei [31], a species' existence is threatened if the water temperature exceeds its upper limit of toleration. Furthermore, according to McDonald et al. [32], lake warming to abovementioned optimum temperatures can result in a reduction in the production of species like lake trout.
In order to explore the infuence of global warming on Limnothrissa miodon population density, we projected lake surface water temperature for the next two decades and used data ftting and numerical simulations to predict the population density of Limnothrissa miodon in Lake Kariba. Our fndings show that the population density of Limnothrissa miodon declines as the lake warms and this is in agreement with Abdellaoui et al. [33], who used a time series analysis  International Journal of Ecology and modelling approach on assessing the impact of temperature and chlorophyll variations on the fuctuations of sardine abundance in Al-Hoceima. Teir results showed an inverse relationship between fuctuations of sardine catch per unit efort and sea surface temperature and that sea surface temperature is the most important parameter affecting the abundance of small pelagic fsh in the Moroccan Mediterranean Sea. Similarly, Lam et al. [34] used simulation models in their study and anticipated that climate change will decrease fsh catches globally by 7.7% by 2050. Our numerical simulation results are in agreement with fndings of the European Environmental Agency's research [35], which showed that lakes are impacted by climate change, mainly by temperature rises [35][36][37]. Our results are congruent with previous studies that have shown signifcant adverse efects of warming on freshwater ecosystems [38][39][40][41]. Our fndings are also supported by Mohammed and Uraguchi [42], who suggested that climate change will have an adverse impact on the already strained fsh resources in sub-Saharan African countries. Te highlights of the study are as follows: (i) A cosine function was ftted to mean lake surface water temperature data and the accuracy of the ftted model was analysed (ii) Default parameter values and diferent initial conditions were used to run numerical simulations for a nutrient, plankton, kapenta, and tigerfsh model with a temperature coefcient for phytoplankton growth (iii) Numerical simulation results of the nonautonomous model showed a stable periodic orbit for varying initial conditions, and therefore, instability  (iv) Lake warming has a negative impact on the more palatable Chlorophyceae, thus leading to decline in kapenta density in Lake Kariba (v) Kapenta population density starts to decline after T opt � 30°C (vi) A decadal warming rate of 0.62°C results in a decline in zooplankton population density, which leads to a decline in the kapenta population density

Conclusions
Numerical results showed that the population density of kapenta declines after the lake surface water temperature surpasses 30°C. Simulation results also showed that lake warming has a negative efect on the more palatable Chlorophyceae population density, resulting in a decline in the density of kapenta in Lake Kariba. Terefore, we conclude that lake surface water temperature infuences the dynamics of Limnothrissa miodon since rising temperatures have been shown to have a negative impact on the kapenta population density in Lake Kariba. Lake surface water temperature infuences the productivity in the water body, altering the mean and amplitude of plankton, kapenta, and tigerfsh population density oscillations. Rising temperatures may lead to the disappearance of the more palatable Chlorophyceae and an abundance of the less palatable Cyanophyceae. Tis may result in a decline in zooplankton populations, and this may adversely afect the kapenta population. Terefore, lake warming may result in a decrease in the kapenta population and reduced kapenta catches in the lake, with the probable consequence of reducing the number of fshing vessels as the fshing will no longer be proftable. Tis can have adverse efects on the local communities and the local economy. In light of these fndings, the governments of Zambia and Zimbabwe and all stakeholders are, therefore, encouraged to take action toward reducing the impacts of climate change in line with the Paris agreement, which was adopted in 2015.
For future studies, we intend to model the dynamics of Limnothrissa miodon with the invasive crayfsh.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.