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This research employs a modified version of the LotkaVolterra nonlinear firstorder ordinary differential equations to model and analyze the parasitic impact of ticks on dogs. The analysis reveals that fluctuations in pesticide effects significantly influence tick populations and the size of the canine host. The study also uncovers that alterations in the size of the interacting species can lead to both stable and unstable states. Interestingly, in a pesticidefree environment, a decline in the intercompetition coefficient catalyzes an increase in the sizes of both interacting species. This increase, although marginal for the tick population, contributes to overall system stability. The findings underscore the utility of the LotkaVolterra nonlinear firstorder ordinary differential equations in modeling the parasitic effect of ticks on dogs. To protect pets, particularly dogs, from the harmful effects of tick infestation, this study recommends the appropriate and regular application of disinfectants.
nonlinear differential equation, LotkaVolterra, system stability, species, dynamical system
A mathematical model serves as a system representation, employing a set of variables, parameters, and equations to delineate relationships between the system components. This approach translates issues from an application area into a comprehensible mathematical formulation, facilitating system explanation, component effect studies, and behavior prediction within the system. Numerous mathematical model types, such as Partial Differential Equations (PDE), Integral Equations, dynamical models, statistical models, ordering differential equations, and functional differential equations, can be used to examine the stability of interactions between ticks and dogs. These models enable the identification, characterization, and comparison of dynamic structures within various natural and artificial systems, seeking to elucidate system behavior [13].
Parasitism is a relationship in which an organism, the parasite, survives at the expense of another organism, the host. Parasitism is a global health issue in animals, primarily resulting from poor hygiene. Parasites, such as mosquitoes, leeches, ticks, hookworms, and lice, are typically significantly smaller than their hosts, do not immediately kill their hosts, and often reside within their hosts for extended periods [46]. Over time, parasites can cause harm to the host, potentially leading to death if not removed.
The LotkaVolterra equations, a pair of firstorder nonlinear differential equations, were designed to model predatorprey relationships and are frequently used to describe the dynamic interactions between predators and prey [79]. This study investigates the parasitic relationship between ticks and dogs. Ticks are ectoparasites that attack the body surface and can transmit diseases to humans and animals alike. The cuticle of hard ticks can expand to accommodate the large volumes of blood ingested, which, in adult ticks, can be anywhere from 200 to 600 times their unfed body weight [10, 11].
This research is of considerable importance to scientists, particularly zoologists and veterinarians, as well as dog owners. It examines the effects of ticks on dogs and offers insights into the parasitic relationship between these two species. Given the public health concerns associated with the spread and control of tickborne diseases, this research is of vital importance. It also investigates a dynamic system of two linear equations that explain tick dynamics to address issues related to the spread of tickborne diseases in infected dogs.
Studies [1214] describe the structure, feeding pattern and the biology of Ticks. While studies [1518] describe the structure, feeding pattern and the biology of dogs.
Research such as those conducted studies [1214] investigate the structure, feeding patterns, and biology of ticks, while others [1518] delve into the same aspects for dogs.
Ticks, the second most prevalent bloodfeeding parasites after mosquitoes [19, 20], destroy blood cells leading to anemia and are carriers of various Protozoa, Viruses, and Bacteria, which can result in tickborne diseases (TBDs) [21]. These diseases encompass both emerging and reemerging infectious diseases. The symptoms of infection typically manifest 27 days post the tick bite. However, the onset of paralysis usually requires multiple simultaneous tick bites. Symptoms in the dog may include hind leg weakness and poor coordination, difficulty swallowing, breathing, and chewing, despite the absence of fever or classic signs of illness. The dog may also appear listless and less mobile. If not promptly addressed, respiratory failure can ensue within hours due to the paralysis of chest muscles.
Experimental studies [2224] have revealed that among the diverse species of ticks infesting dogs, the brown tick (Rhipicephalus sanguineus) is the most widespread. Other relevant works [2529]. Studies relating to the impact of ticks on dogs can be found in the studies [3034]. Opanuga et al. [35] and Edeki et al. [36] provided the underlying differential equations which is useful in the current study. Studies [2739] relate to tickborne infectious diseases affecting dogs other related studies [4043]. Another nonlinear differential approach was provided by Adesina et al. [44]. Various studies on dogs and human tickborne infections can be found in studies [4550]. Agboola et al. [51] presented the solution of third order ordinary differential equations using differential transform method which is relevant to the current study.
The objective of this study is to delineate the relationship between two biological species, ticks and dogs, utilizing a numerical computational scheme predicated on the LotkaVolterra nonlinear firstorder ordinary differential equations. Specifically, this research aims to (i) assess the parasitic effect of ticks on dogs, (ii) evaluate the influence of pesticides on system stability, and (iii) analyze the impact of the dog's intercompetition coefficient on the system.
This study intends to augment the existing body of literature in mathematical modeling and computational mathematics, providing insights into the relationship between these two biological species. It seeks to elucidate the mutual effects of these species on each other and the overall impact of the parasite (ticks) on the host (dogs). Additionally, it aims to guide scientists in monitoring the survival of biological species.
2.1 Model formation
Considering the relationship between two biological species where one of the species N_{1} (ticks) depend on the other species N_{2}(dog), the modified system of LotkaVolterra nonlinear first order ordinary differential equations of the form of the LotkaVolterra logistic model is considered as given [52]:
$\begin{gathered}\frac{d N_1}{d t}=a_1 N_1a_2 N_1^2+\alpha N_1 N_2\rho_1 N_1 \\ N 1(0)=N 10 \geq 0\end{gathered}$ (1)
$\begin{gathered}\frac{d N_2}{d t}=b_1 N_2b_2 N_2^2\beta N_1 N_2 \\ N 2(0)=N 20 \geq 0\end{gathered}$ (2)
2.2 Mathematical formulation
Considering the two biological species of LotkaVolterra logistic model with one species obtaining resource from the other, this situation leads to a relationship between the species causing both species to experience a parasitic interaction. The system above can be clearly explained using nonlinear first order differential equation. The parameters in the model are contained in the governing pair of firstorder nonlinear differential equations. The parameters sufficiently explain the preypredator interactions. The parameters are defined as follows:
$N_1$ is the population size of the first species (ticks).
$N_2$ is the population size of the second species (dog).
$a_1$ is the intrinsic growth rate of the first species.
$a_2$ is the intracompetition coefficient of the first species.
$b_1$ is the intrinsic effect on the second species.
$b_2$ is the intracompetition coefficient of the second species.
α is the intercompetition coefficient of the first species.
β is the inter competition coefficient of the second species.
$\rho_1$ is the pesticide to inhibit the growth of $N_1$.
It is imperative to note that both Eqs. (1)(2) conform with the logistic equation whereby the tick species affects the growth of the second species growth through the parasitic relationship that exist between the two species. ρ_{1} represents a control mechanism to inhibit the excessive growth of the first species.
2.3 Determination of the steady state solution
A system is said to reach a steady state or equilibrium when it exhibits no further tendency to change its property over time. That is, if the system is in a steadystate at time to then it will stay there for all times $t \geq t_0$. A detailed definition and mathematical analysis of the concept of steadystate and its stability is reported [5254]. According to linear stability analysis, a steadystate solution is stable if all the Eigen values of the Jacobins matrix evaluated at that steady state solution have negative real parts. The study [55] is a related ordinary differential equations approach.
$\frac{d N_1}{d t}=\frac{d N_2}{d t}=0$
For Eq. (1),
$\frac{d N_1}{d t}=a_1 N_1a_2 N_1^2+\alpha N_1 N_2\alpha_1 N_1$ (3)
Again, from Eq. (2),
$\frac{d N_2}{d t}=b_1 N_2b_2 N_2^2+\beta N_1 N_2$ (4)
Since the righthand side of the equation is not equal to zero, Eq. (1) gives:
$\left.\begin{array}{c}a_1 N_1a_2 N_1^2+\alpha N_1 N_2\alpha_1 N_1=0 \\ N_2\left(a_1a_2 N_1+a N_2a_1\right)=0 \\ N_1=0 \text { or } N_1=\frac{1}{a_2}\left(a_1+a N_2a_1\right)\end{array}\right\}$ (5)
Similarly, Eq. (2) gives:
$\left.\begin{array}{c}b_1 N_2b_2 N_2^2\beta N_1 N_2=0 \\ N_2\left(b_1b_2 N_2\beta N_1\right)=0 \\ N_2=0 \text { or } N_2=\frac{1}{b_2}\left(b_1\beta N_1\right)\end{array}\right\}$ (6)
Thus, when N_{1}=0 and N_{2}=0 is the point (0, 0) which is the trival steady state solution. This implies that both species have gone into extinction.
For N_{1}=0 and N_{2}≠0, then $N_2=\frac{1}{b_2}\left(b_1+\rho_1\right)=N_2^*$, therefore (0, $N_2^*$) is a steady state solution where the second species (Dog) has not been infested yet.
For N_{1}≠0 and N_{2}=0, then $N_1=\frac{1}{a_2}\left(a_1\alpha_1\right)=N_1^*$, also, the above expression gives ($N_1^*$, 0), which is a steadystate solution where the first species (Ticks) is healthy and the second species has been infested.
For N_{1}≠0 and N_{2}≠0, then $N_1=\frac{1}{a_2}\left(a_1+\alpha N_2\alpha_1\right)$,
$\begin{gathered}N_1=\frac{1}{a_2}\left[a_1+\alpha\left(\frac{1}{b_2}\left(b_1\beta N_1\right)\right)\alpha_1\right] \\ =\frac{1}{a_2}\left[a_1\frac{\alpha b_1}{b_2}\frac{\alpha b_1 N_1}{b_1}\alpha_1\right] \\ a_2 N_1=a_1\frac{\alpha b_1}{b_2}\frac{\alpha b_1 N_1}{b_2}\alpha_1 \\ a_2 N_1+\frac{\alpha \beta N_1}{b_2}=\frac{b_2 a_1\alpha b_1\alpha_1 b_2}{b_2} \\ \frac{a_2 b_2 N_2+\beta N_1}{b_2}=\frac{b_2 a_1\alpha b_1\alpha_1 b_2}{b_2} \\ N_1\left(a_2 b_2\alpha \beta\right)=a_1 b_2\alpha b_1\alpha_1 b_2\end{gathered}$
$N_1=\frac{1}{a_2 b_2+\alpha \beta}\left(a_1 b_2\alpha b_1\alpha_1 b_2\right)=N_1^{* *}$ (7)
Similarly,
$\begin{gathered}N_2=\frac{1}{b_2}\left(b_1\beta N_1\right) \\ \frac{1}{b_2}\left[b_1+\beta\left(\frac{1}{a_2}\left(a_1+\alpha N_2\alpha_1\right)\right)\right] \\ \frac{1}{b 2}\left[b_1\frac{\beta a_1}{a_2}\frac{\alpha \beta N_2}{a_2}+\frac{\beta \alpha_1}{a_2}\right] \\ b_2 N_2+\frac{\alpha \beta N_2}{a_2}=b_1\frac{\beta a_1}{a_2}+\frac{\beta \alpha_1}{a_2} \\ a_2 b_2 N_2+\alpha \beta N_2=a_2 b_1\beta a_1+\beta \alpha_1 \\ N_2\left(a_2 b_2+\alpha \beta\right)=a_2 b_1\beta a_1+\beta \alpha_1\end{gathered}$
$N_2=\frac{1}{a_2 b_2+\alpha \beta}\left(a_2 b_1\beta a_1+\alpha \beta\right)=N_2^{* *}$ (8)
At this point ($N_1^{* *}$, $N_2^{* *}$), there is a coexistence of both species.
2.3.1 Characterization of the steady state solution of the interacting function
In characterization of the steady state solution, steady state equation is generalized using state variables in order to obtain Jacobian Matrix elements as given by:
$a_1 N_1a_2 N_1^2+\alpha N_1 N_2\alpha_1 N_1=0$
Let,
$f\left(N_1, N_2\right)=a_1 N_1a_2 N_1^2\alpha N_1 N_2\alpha_1 N_1$
And,
$f\left(N_1, N_2\right)=b_1 N_2b_2 N_2^2\beta N_1 N_2$ (9)
N_{1} and N_{2} is this instance are state variables. Differentiating the above equations with respect to state variables to obtain Jacobian elements gives:
$\begin{gathered}J_{11}=\frac{\partial y}{\partial N_1}=a_12 a_2 N_1+\alpha N_2\alpha_1 \\ J_{12}=\frac{\partial y}{\partial N_2}=\alpha N_1 \\ J_{21}=\frac{\partial y}{\partial N_1}=\beta N_2 \\ J_{22}=\frac{\partial y}{\partial N_2}=b_12 b_2 N_2+\beta N_2\beta N_1\end{gathered}$
At the trivial steady state solution (0, 0),
$\begin{gathered}J_{11}=a_12 a_2(0)+\alpha(0)\alpha_1=a_1\alpha_1 \\ J_{12}=\alpha(o)=0 \\ J_{21}=\beta(0)=0 \\ J_{22}=b_12 b_2(0)+\beta(0)\beta(0)=b_1\end{gathered}$
The Jacobian matrix becomes,
$J_1=\left[\begin{array}{cc}a_1\alpha_1 & 0 \\ 0 & b_1\end{array}\right]$ and $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
The characteristic equation is,
$\begin{gathered}\operatorname{Det}\left(J_1\lambda I\right)=0 \\ \left\begin{array}{cc}a_1\alpha_1\lambda & 0 \\ 0 & b_1\lambda\end{array}\right=0 \\ a_1\alpha_1\lambda=0 \text { and } b_1\lambda=0 \\ \lambda_1=a_1\alpha \text { and } \lambda_2=b_1\end{gathered}$
Therefore, $\lambda_1=a_1\alpha_1$ and $\lambda_2=b_1$ are the eigenvalues. The trivial steady state solution is unstable since both eigenvalues are positive.
At the trivial steady state $\left(0, \frac{b_1}{b_2}\right)$,
$\begin{gathered}J_{11}=a_12 a_2 N_1+\rho N_2\rho_1 \\ a_1+\rho N_2\rho_1 \\ a_1+\rho\left(\frac{b_1}{b_2}\right)\rho_1 \\ a_1+\frac{\rho b_1}{b_2}\rho_1 \\ J_{12}=\rho N_1=0 \\ J_{21}=\beta N_2=\beta\left(\frac{b_1}{b_2}\right)=\left(\frac{\beta b_1}{b_2}\right) \\ J_{22}=b_12 b_2 N_2+\beta N_2\beta N_1=b_12 b_2\left(\frac{b_1}{b_2}\right) \\ =b_12 b_1=b_1\end{gathered}$
The Jacobian matrix is,
$J_2=\left[\begin{array}{cc}a_1\frac{\alpha b_1}{b_2} & 0 \\ \frac{\beta b_1}{b_2} & b_1\end{array}\right]$
The characteristic equation is,
$\begin{gathered}\operatorname{Det}\left(J_2\lambda I\right)=\left\begin{array}{cc}a_1\frac{\alpha b_1}{b_2}\rho_1\lambda & 0 \\ \frac{\beta b_1}{b_2} & b_1\lambda\end{array}\right=0 \\ a_1\frac{\alpha b_1}{b_2}\rho_1\lambda=0\end{gathered}$
$\lambda_1=a_1\frac{\rho b_1}{b_2}\rho_1$ (10)
And,
$\begin{gathered}b_1\lambda=0 \\ \lambda_2=b_1\end{gathered}$ (11)
The steady state at $\left(0, \frac{b_1}{b_2}\right)$ is unstable since the eigenvalues are positive and negative.
At the trivial steady state $\left(\frac{a_1\rho_1}{a_2}, 0\right)$,
$\begin{gathered}J_{11}=a_12 a_2\left[\frac{a_1\rho_1}{a_2}\right]+\rho(0)\rho_1 \\ =a_12 a_1+2 \rho_1\rho_1 \\ =a_1+\rho_1 \\ J_{12}=\rho\left[\frac{a_1\rho_1}{a_2}\right] \\ J_{21}=\beta(0)=0 \\ J_{22}=b_12 b_2(0)+\beta\left[\frac{a_1\rho_1}{a_2}\right] \\ b_1\beta\left[\frac{a_1\rho_1}{a_2}\right]\end{gathered}$
The Jacobian matrix is:
$J_3=\left[\begin{array}{cc}\rho_1a_1 & \rho\left(\frac{a_1\alpha_1}{a_2}\right) \\ 0 & b_1\beta\left(\frac{a_1\rho}{a_2}\right)\end{array}\right]$
The characteristic equation is:
$\begin{aligned} \operatorname{Det}\left(J_3\lambda I\right) & =\left\begin{array}{cc}\alpha_1a_1\lambda & \rho\left(\frac{a_1\alpha_1}{a_2}\right) \\ 0 & b_1\beta\left(\frac{a_1\alpha_1}{a_2}\right)\lambda\end{array}\right=0 \\ & =\rho_1a_1\lambda=0, \lambda_1=\rho_1a_1\end{aligned}$
And,
$\lambda_2=b_1\beta\left[\frac{a_1\alpha_1}{a_2}\right]\lambda$ (12)
Considering the eigenvalues which are positive, this means that the steady state solution is unstable.
2.4 Method of solution
The numerical simulation was conducted using MATLAB software and the programming language provided in the package (oDE45) with reference to the numerical system of Eqs. (1)(2). Following the procedure outlined [52], the following parameters were obtained a_{1}=5, a_{2}=0.22, α=0.007, b_{1}=3, b_{2}=0.26, β=0.008 while the values of ρ_{1}=3.5, β_{1}=1.4 are randomly selected. N_{1}and N_{2} are obtained based on Eqs. (7)(8), λ_{1} and λ_{2} are obtained based on Eq. (10) and Eq. (12) respectively. We present the simulation scheme based on the Eqs. (1)(12) in Table 1, as follows:
Table 1. Simulation scheme
Case 
Effect 
On 
1 
$\Delta \rho_1$ 
N_{1}, N_{2} 
2 
$+\Delta \rho_1 \mathrm{n}$ 
N_{1}, N_{2} 
3 
$\Delta \rho_1$ 
$\lambda_1$ and $\lambda_2$ 
4 
$+\Delta \rho_1$ 
$\lambda_1$ and $\lambda_2$ 
5 
$\Delta \mathrm{N}_1$ 
$\lambda_1$ and $\lambda_2$ 
6 
$\Delta \mathrm{N}_2$ 
$\lambda_1$ and $\lambda_2$ 
7 
$\Delta \mathrm{N}_1$ and $\Delta \mathrm{N}_2$ 
$\lambda_1$ and $\lambda_2$ 
8 
β 
N_{1}, N_{2} 

β 
$\lambda_1$ and $\lambda_2$ 
where,
$\Delta \rho_1$ is the decrease in pesticides
$+\Delta \rho_1$ is the increase in pesticides
$\Delta \mathrm{N}_1$ is decrease in ticks population
$\Delta \mathrm{N}_2$ is decrease in ticks population
β is intercompetition of the 2^{nd} species
Given the parameters, the study seeks to obtain the results outlined in Table 1 as follows:
(i) the impact of decrease in pesticide $\Delta \rho_1$ on ticks size of ticks, N_{1} and dog size N_{2} is sought.
(ii) the impact of increase in pesticide, $+\Delta \rho_1$ , on ticks size of ticks, N_{1} and dog size N_{2} is sought
(iii) the impact of ($\Delta \rho_1$) on tick on the stability $\lambda_1$ and $\lambda_2$ of the system.
(iv) the impact of ($+\Delta \rho_1$) on tick on the stability $\lambda_1$ and $\lambda_2$ of the system is sought.
(v) the effect of decreasing the tick’s population ($\Delta \mathrm{N}_1$) on the stability $\lambda_1$ and $\lambda_2$ of the system.
(vi) the effect of decreasing the dog’s population($\Delta \mathrm{N}_2$) on the stability$\lambda_1$ and $\lambda_2$ of the system is sought.
(vii) the effect of simultaneously decreasing the population of both species ($\Delta \mathrm{N}_1$ and $\Delta \mathrm{N}_2$) on the stability $\lambda_1$ and $\lambda_2$ of the system is sought.
(viii) the effect of the intercompetition of the 2nd species (β), on the population of competing species, N_{1} and N_{2}.
(ix) the effect of the intercompetition of the 2nd species (β), on the stability $\lambda_1$ and $\lambda_2$ of the system is sought.
Table 2 shows that as the volume of pesticide increase, the number of ticks increase, and the number of dogs increases. By implication, the mortality rate of dogs decreases.
Table 3 shows that the increase in volume of pesticides, results in a significant decrease in the size of the ticks, and a resultant gradual increase in the size of the dogs.
Table 4 shows that decreasing the impact of pesticide on ticks’ results in a stable dynamical system, by implication, there wouldn’t be increase without bound in the number either dog or tick in a given dynamical ecological system.
Table 2. Impact of decreasing the effects of pesticides, ρ_{1}, on the populations of competing species, N_{1} and N_{2}
$+\Delta \rho_1$ 
N_{1} 
N_{2} 
3.5 
7.1783 
11.3167 
3.3250 
7.9730 
11.2921 
3.1500 
8.7677 
11.2675 
2.9750 
9.5624 
11.2430 
2.8000 
10.3572 
11.2184 
2.6250 
11.1521 
11.1930 
2.4500 
11.9470 
11.1688 
2.2750 
12.7430 
11.1451 
2.1000 
13.5447 
11.1282 
1.9250 
14.3316 
11.0980 
1.7500 
15.1297 
11.0741 
1.5750 
15.9203 
11.0487 
1.4000 
16.7150 
11.0242 
1.2250 
17.5103 
10.9998 
1.0500 
18.3059 
10.9754 
0.8750 
19.0996 
10.9509 
0.7000 
19.8946 
10.9265 
0.5250 
20.6899 
10.9020 
0.3500 
21.4857 
10.8776 
0.1750 
22.2805 
10.8532 
Table 3. Impact of increasing the effects of pesticides, ρ_{1}, on the populations of competing species, N_{1} and N_{2}
$+\Delta \rho_1$ 
N_{1} 
N_{2} 
3.5 
7.1783 
11.3167 
3.6756 
6.3836 
11.3413 
3.8500 
5.5890 
11.3658 
4.0250 
4.7943 
11.3903 
4.3750 
3.2049 
11.4392 
4.5500 
2.4101 
11.4636 
4.7250 
1.6144 
11.4881 
4.9000 
0.8218 
11.5124 
5.0750 
0.1925 
11.5316 
5.2500 
0.0140 
11.5371 
5.4250 
0.0006 
11.5376 
5.6000 
1.9570×10^{5} 
11.5376 
5.7750 
6.3650×10^{7} 
11.5369 
5.9500 
2.0995×10^{8} 
11.5382 
6.1250 
1.0935×10^{9} 
11.5379 
6.3000 
1.4951×10^{10} 
11.5371 
6.4750 
1.2559× 10^{11} 
11.5355 
6.6500 
1.4784×10^{11} 
11.5362 
6.8250 
1.1880×10^{12} 
11.5348 
7.0000 
6.4500×10^{13} 
11.5378 
Table 4. Impact of decreasing the effects of pesticides, ρ_{1}, on the stability of the system (ToS)
$+\Delta \rho_1$ 
$\lambda_1$ 
$\lambda_2$ 
ToS 
3.5 
1.7415 
2.9383 
Stable 
3.3250 
1.9171 
2.9307 
Stable 
3.1500 
2.0933 
2.9226 
Stable 
2.9750 
2.2704 
2.9136 
Stable 
2.8000 
2.4496 
2.9025 
Stable 
2.6250 
2.6358 
2.8841 
Stable 
2.4500 
2.8441 
2.8441 
Stable 
2.2750 
2.9287 
2.9287 
Stable 
2.1000 
3.0954 
2.9371 
Stable 
1.9250 
3.2864 
2.9078 
Stable 
1.7500 
3.4686 
2.8955 
Stable 
1.5750 
3.6445 
2.8855 
Stable 
1.4000 
3.8209 
2.8771 
Stable 
1.2250 
3.9971 
2.8695 
Stable 
1.0500 
4.1729 
2.8622 
Stable 
0.8750 
4.3477 
2.8551 
Stable 
0.7000 
4.5229 
2.8481 
Stable 
0.5250 
4.6981 
2.8413 
Stable 
0.3500 
4.8734 
2.8346 
Stable 
0.1750 
5.0483 
2.8280 
Stable 
Table 5 is a replica of Table 4, which shows that increasing the impact of pesticide on ticks’ results in a stable dynamical system, by implication, there wouldn’t be increase without bound in the number either dog or tick in a given dynamical ecological system. This shows that variations in the effects of pesticide while other model parameters are fixed results in a stable system.
Table 6 shows that as N_{1} decreases, the dynamical system is stable to a point, util it gets to a point when it becomes progressively unstable as N_{1} further approach zero.
Table 7 shows that as N_{1 }decreases, the dynamical system is stable to a point, until it gets to a point when it becomes progressively unstable as N_{1} further approach zero.
In Table 8, a simultaneous decrease in the size of interacting species N_{1} and N_{2}, the dynamical system is stable to a point, until it gets to a point when it becomes progressively unstable as N_{1} further approach zero.
Table 5. Impact of increasing the effects of pesticides, ρ_{1}, on the stability of the system
$+\Delta \rho_1$ 
$\lambda_1$ 
$\lambda_2$ 
ToS 
3.5 
1.7415 
2.9383 
Stable 
3.6750 
1.5661 
2.9456 
Stable 
3.8500 
1.3910 
2.9526 
Stable 
4.0250 
1.2160 
2.9595 
Stable 
4.3750 
0.8662 
2.9730 
Stable 
4.5500 
0.6914 
2.9797 
Stable 
4.7250 
0.5162 
2.9863 
Stable 
4.9000 
0.3424 
2.9928 
Stable 
5.0750 
0.2404 
2.9979 
Stable 
5.2500 
0.3369 
2.9994 
Stable 
5.4250 
0.5060 
2.9995 
Stable 
5.6000 
0.6808 
2.9995 
Stable 
5.7750 
0.8558 
2.9992 
Stable 
5.9500 
1.0308 
2.9998 
Stable 
6.1250 
1.2058 
3.0000 
Stable 
6.3000 
1.3808 
2.9993 
Stable 
6.4750 
1.5557 
2.9985 
Stable 
6.6500 
1.7308 
2.9988 
Stable 
6.8250 
1.9057 
2.9981 
Stable 
7.0000 
2.0808 
2.9996 
Stable 
Table 6. The effect of decreasing the population of ticks on the stability of the system
$\Delta N_1$ 
$\lambda_1$ 
$\lambda_2$ 
ToS 
7.1783 
1.7415 
2.9383 
Stable 
6.8194 
1.5829 
2.9361 
Stable 
6.4605 
1.4245 
2.9337 
Stable 
6.1016 
1.2662 
2.9312 
Stable 
5.7426 
1.1080 
2.9286 
Stable 
5.3837 
0.9498 
2.9260 
Stable 
5.0245 
0.7916 
2.9234 
Stable 
4.6659 
0.6335 
2.9207 
Stable 
4.3070 
0.4754 
2.9180 
Stable 
3.9481 
0.3173 
2.9153 
Stable 
3.5892 
0.1593 
2.9126 
Stable 
3.2302 
0.0012 
2.9298 
Stable 
2.8713 
0.1568 
2.9071 
Unstable 
2.5124 
0.3148 
2.9043 
Unstable 
2.1535 
0.4728 
2.9015 
Unstable 
1.7946 
0.6308 
2.8987 
Unstable 
1.4357 
0.7888 
2.8959 
Unstable 
1.0767 
0.9468 
2.8931 
Unstable 
0.7178 
1.1048 
2.8903 
Unstable 
0.3589 
1.2628 
2.8875 
Unstable 
Table 7. The effect of decreasing the population of Dog, N_{2}, on the stability of the system
$\Delta N_2$ 
$\lambda_1$ 
$\lambda_2$ 
ToS 
11.3167 
1.7415 
2.9383 
Stable 
10.7509 
1.7385 
2.6431 
Stable 
10.1850 
1.7364 
2.3470 
Stable 
9.6192 
1.7378 
2.0474 
Stable 
9.0534 
1.7435 
1.7435 
Stable 
8.4875 
1.7032 
1.4856 
Stable 
7.9217 
1.7079 
1.1827 
Stable 
7.3559 
1.7064 
0.8861 
Stable 
6.7900 
1.7035 
0.5907 
Stable 
6.2242 
1.7002 
0.2958 
Stable 
5.6584 
1.6967 
0.0011 
Stable 
5.0925 
1.6931 
0.2934 
Unstable 
4.5267 
1.6893 
0.5879 
Unstable 
3.9608 
1.6856 
0.8823 
Unstable 
3.3950 
1.6817 
1.1767 
Unstable 
2.8292 
1.6779 
1.4710 
Unstable 
2.2633 
1.6740 
1.7654 
Unstable 
1.6975 
1.6702 
2.0597 
Unstable 
1.1317 
1.6663 
2.3540 
Unstable 
0.5658 
1.6624 
2.6483 
Unstable 
Table 8. The effect of simultaneously decreasing the population of both species, N_{1} and N_{2}, on the stability of the system
$\Delta N_1$ 
$\lambda_1$ 
$\lambda_2$ 
ToS 
ToS 
7.1783 
11.3167 
1.7415 
2.9383 
Stable 
6.8194 
10.750 
1.5796 
2.6412 
Stable 
6.4605 
10.185 
1.4179 
2.3439 
Stable 
6.1016 
9.6192 
1.2562 
2.0467 
Stable 
5.7426 
9.0534 
1.0946 
1.7493 
Stable 
5.3837 
8.4875 
0.9331 
1.4517 
Stable 
5.0248 
7.9217 
0.7721 
1.1537 
Stable 
4.6659 
7.3559 
0.6122 
0.8547 
Stable 
4.3070 
6.7900 
0.4578 
0.5500 
Stable 
3.9481 
6.2242 
0.2744 
0.2843 
Stable 
3.5892 
5.6584 
0.1107 
0.0208 
Unstable 
3.2302 
5.0925 
0.0463 
0.3228 
Unstable 
2.8713 
4.5267 
0.2067 
0.6214 
Unstable 
2.5124 
3.9608 
0.3678 
0.9193 
Unstable 
2.1535 
3.3950 
0.5293 
1.2168 
Unstable 
1.7946 
2.8292 
0.6909 
1.5141 
Unstable 
1.4357 
2.2633 
0.8527 
1.8114 
Unstable 
1.0767 
1.6975 
1.0144 
2.1086 
Unstable 
0.7178 
1.1317 
1.1763 
2.4058 
Unstable 
0.3589 
0.5658 
1.3381 
2.7029 
Unstable 
Table 9 shows that in other a decrease in β, increases the population of both species, with the increment more significant in N_{2}.
Table 10 shows that a decrease in β, results in a stable system as both eigenvalues, $\lambda_1$ and $\lambda_2$, are negative. According to the linear stability analysis a dynamical system is stable if all the Eigen values of the Jacobian matrix are negative. But if one of the Eigen values is positive the system is unstable.
Table 9. Evaluating the effect of the intercompetition of the 2^{nd} species (β), on the population of competing species, N_{1} and N_{2}
β 
N_{1} 
N_{2} 
0.0080 
23.0841 
10.8290 
0.0076 
23.0854 
10.8645 
0.0072 
23.0866 
10.8999 
0.0068 
23.0878 
10.9354 
0.0064 
23.0890 
10.9708 
0.0060 
23.0816 
11.0060 
0.0056 
23.0829 
11.0415 
0.0052 
23.0841 
11.0770 
0.0048 
23.0845 
11.1125 
0.0044 
23.0867 
11.1479 
0.0040 
23.0879 
11.1834 
0.0036 
23.0892 
11.2189 
0.0032 
23.0905 
11.2544 
0.0028 
23.0917 
11.2899 
0.0024 
23.0930 
11.3254 
0.0020 
23.0943 
11.3609 
0.0016 
23.0956 
11.3964 
0.0012 
23.0969 
11.4319 
0.0008 
23.0982 
11.4674 
0.0004 
23.0995 
11.5029 
Table 10. Evaluating the effect of the intercompetition of the 2^{nd} species (β), on the stability of the system
β 
$\lambda_1$ 
$\lambda_2$ 
ToS 
0.0080 
5.2270 
2.8216 
Stable 
0.0076 
5.2281 
2.8305 
Stable 
0.0072 
5.2291 
2.8395 
Stable 
0.0068 
5.2301 
2.8484 
Stable 
0.0064 
5.2312 
2.8574 
Stable 
0.0060 
5.2284 
2.8661 
Stable 
0.0056 
5.2295 
2.8751 
Stable 
0.0052 
5.2306 
2.8840 
Stable 
0.0048 
5.2317 
2.8930 
Stable 
0.0044 
5.2328 
2.9019 
Stable 
0.0040 
5.2339 
2.9108 
Stable 
0.0036 
5.2350 
2.9198 
Stable 
0.0032 
5.2361 
2.9287 
Stable 
0.0028 
5.2372 
2.9376 
Stable 
0.0024 
5.2383 
2.9466 
Stable 
0.0020 
5.2394 
2.9555 
Stable 
0.0016 
5.2405 
2.9644 
Stable 
0.0012 
5.2417 
2.9733 
Stable 
0.0008 
5.2428 
2.9935 
Stable 
0.0004 
5.2440 
2.9911 
Stable 
This study underscores the importance of prompt tick identification and treatment in dogs, bringing to light the severe consequences of unchecked tick infestations. Utilizing a system of nonlinear firstorder differential equations, we explored the intricate dynamics between these two biological species.
For future research, we recommend an extension of this work using a system of secondorder differential equations. This could potentially provide deeper insights into the more complex interactions and dynamics that characterize this parasitic relationship. Beyond this, there may be a wealth of other parameters, such as environmental factors, the host's health status, or the specific species of ticks involved, that could influence the population dynamics of the interacting species. These parameters could be the focus of future investigations.
Moreover, exploring the effects of the competition coefficient on the populations of biological species might provide valuable information. For example, how does the presence of other parasites or potential hosts in the environment influence the tickdog interaction? Could a higher competition coefficient lead to a decrease in tick populations, thereby reducing the risk for dogs?
Finally, future studies may consider conducting clinical trials to validate and extend the findings of this study. Realworld testing could provide a more comprehensive understanding of the practical implications of our theoretical models, helping to bridge the gap between mathematical modeling and veterinary practice.
In conclusion, this study contributes to the existing body of knowledge by shedding light on the adverse effects of tick infestations in dogs and offering a mathematical model to understand the dynamics of such parasitic relationships. We believe the pathways we have highlighted for future research will pave the way for more comprehensive investigations, ultimately benefiting both veterinary science and the welfare of animals.
The authors thank Covenant University Centre for Research, Innovation, and Discovery (CUCRID) for their support in making this research a reality.
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