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This research employs a modified version of the Lotka-Volterra non-linear first-order 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 pesticide-free environment, a decline in the inter-competition 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 Lotka-Volterra non-linear first-order 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.
non-linear differential equation, Lotka-Volterra, 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 [1-3].
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 [4-6]. Over time, parasites can cause harm to the host, potentially leading to death if not removed.
The Lotka-Volterra equations, a pair of first-order nonlinear differential equations, were designed to model predator-prey relationships and are frequently used to describe the dynamic interactions between predators and prey [7-9]. 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 tick-borne 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 tick-borne diseases in infected dogs.
Studies [12-14] describe the structure, feeding pattern and the biology of Ticks. While studies [15-18] describe the structure, feeding pattern and the biology of dogs.
Research such as those conducted studies [12-14] investigate the structure, feeding patterns, and biology of ticks, while others [15-18] delve into the same aspects for dogs.
Ticks, the second most prevalent blood-feeding parasites after mosquitoes [19, 20], destroy blood cells leading to anemia and are carriers of various Protozoa, Viruses, and Bacteria, which can result in tick-borne diseases (TBDs) [21]. These diseases encompass both emerging and re-emerging infectious diseases. The symptoms of infection typically manifest 2-7 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 [22-24] have revealed that among the diverse species of ticks infesting dogs, the brown tick (Rhipicephalus sanguineus) is the most widespread. Other relevant works [25-29]. Studies relating to the impact of ticks on dogs can be found in the studies [30-34]. Opanuga et al. [35] and Edeki et al. [36] provided the underlying differential equations which is useful in the current study. Studies [27-39] relate to tick-borne infectious diseases affecting dogs other related studies [40-43]. Another non-linear differential approach was provided by Adesina et al. [44]. Various studies on dogs and human tick-borne infections can be found in studies [45-50]. 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 Lotka-Volterra non-linear first-order 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 inter-competition 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 N1 (ticks) depend on the other species N2(dog), the modified system of Lotka-Volterra non-linear first order ordinary differential equations of the form of the Lotka-Volterra logistic model is considered as given [52]:
$\begin{gathered}\frac{d N_1}{d t}=a_1 N_1-a_2 N_1^2+\alpha N_1 N_2-\rho_1 N_1 \\ N 1(0)=N 10 \geq 0\end{gathered}$ (1)
$\begin{gathered}\frac{d N_2}{d t}=b_1 N_2-b_2 N_2^2-\beta N_1 N_2 \\ N 2(0)=N 20 \geq 0\end{gathered}$ (2)
2.2 Mathematical formulation
Considering the two biological species of Lotka-Volterra 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 non-linear first order differential equation. The parameters in the model are contained in the governing pair of first-order nonlinear differential equations. The parameters sufficiently explain the prey-predator 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 intra-competition coefficient of the first species.
$b_1$ is the intrinsic effect on the second species.
$b_2$ is the intra-competition coefficient of the second species.
α is the inter-competition co-efficient 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. ρ1 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 steady-state 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 steady-state and its stability is reported [52-54]. According to linear stability analysis, a steady-state 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),
$\frac{d N_1}{d t}=a_1 N_1-a_2 N_1^2+\alpha N_1 N_2-\alpha_1 N_1$ (3)
Again, from Eq. (2),
$\frac{d N_2}{d t}=b_1 N_2-b_2 N_2^2+\beta N_1 N_2$ (4)
Since the right-hand side of the equation is not equal to zero, Eq. (1) gives:
$\left.\begin{array}{c}a_1 N_1-a_2 N_1^2+\alpha N_1 N_2-\alpha_1 N_1=0 \\ N_2\left(a_1-a_2 N_1+a N_2-a_1\right)=0 \\ N_1=0 \text { or } N_1=\frac{1}{a_2}\left(a_1+a N_2-a_1\right)\end{array}\right\}$ (5)
Similarly, Eq. (2) gives:
$\left.\begin{array}{c}b_1 N_2-b_2 N_2^2-\beta N_1 N_2=0 \\ N_2\left(b_1-b_2 N_2-\beta N_1\right)=0 \\ N_2=0 \text { or } N_2=\frac{1}{b_2}\left(b_1-\beta N_1\right)\end{array}\right\}$ (6)
Thus, when N1=0 and N2=0 is the point (0, 0) which is the trival steady state solution. This implies that both species have gone into extinction.
For N1=0 and N2≠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 N1≠0 and N2=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 steady-state solution where the first species (Ticks) is healthy and the second species has been infested.
For N1≠0 and N2≠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 co-existence 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:
$a_1 N_1-a_2 N_1^2+\alpha N_1 N_2-\alpha_1 N_1=0$
Let,
$f\left(N_1, N_2\right)=a_1 N_1-a_2 N_1^2-\alpha N_1 N_2-\alpha_1 N_1$
And,
$f\left(N_1, N_2\right)=b_1 N_2-b_2 N_2^2-\beta N_1 N_2$ (9)
N1 and N2 is this instance are state variables. Differentiating the above equations with respect to state variables to obtain Jacobian elements gives:
$\begin{gathered}J_{11}=\frac{\partial y}{\partial N_1}=a_1-2 a_2 N_1+\alpha N_2-\alpha_1 \\ J_{12}=\frac{\partial y}{\partial N_2}=\alpha N_1 \\ J_{21}=\frac{\partial y}{\partial N_1}=-\beta N_2 \\ J_{22}=\frac{\partial y}{\partial N_2}=b_1-2 b_2 N_2+\beta N_2-\beta N_1\end{gathered}$
At the trivial steady state solution (0, 0),
$\begin{gathered}J_{11}=a_1-2 a_2(0)+\alpha(0)-\alpha_1=a_1-\alpha_1 \\ J_{12}=\alpha(o)=0 \\ J_{21}=-\beta(0)=0 \\ J_{22}=b_1-2 b_2(0)+\beta(0)-\beta(0)=b_1\end{gathered}$
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 $\left(0, \frac{b_1}{b_2}\right)$,
$\begin{gathered}J_{11}=a_1-2 a_2 N_1+\rho N_2-\rho_1 \\ a_1+\rho N_2-\rho_1 \\ a_1+\rho\left(\frac{b_1}{b_2}\right)-\rho_1 \\ a_1+\frac{\rho b_1}{b_2}-\rho_1 \\ J_{12}=\rho N_1=0 \\ J_{21}=-\beta N_2=-\beta\left(\frac{b_1}{b_2}\right)=\left(\frac{-\beta b_1}{b_2}\right) \\ J_{22}=b_1-2 b_2 N_2+\beta N_2-\beta N_1=b_1-2 b_2\left(\frac{b_1}{b_2}\right) \\ =b_1-2 b_1=-b_1\end{gathered}$
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 $\left(\frac{a_1-\rho_1}{a_2}, 0\right)$,
$\begin{gathered}J_{11}=a_1-2 a_2\left[\frac{a_1-\rho_1}{a_2}\right]+\rho(0)-\rho_1 \\ =a_1-2 a_1+2 \rho_1-\rho_1 \\ =-a_1+\rho_1 \\ J_{12}=\rho\left[\frac{a_1-\rho_1}{a_2}\right] \\ J_{21}=-\beta(0)=0 \\ J_{22}=b_1-2 b_2(0)+\beta\left[\frac{a_1-\rho_1}{a_2}\right] \\ b_1-\beta\left[\frac{a_1-\rho_1}{a_2}\right]\end{gathered}$
The Jacobian matrix is:
$J_3=\left[\begin{array}{cc}\rho_1-a_1 & \rho\left(\frac{a_1-\alpha_1}{a_2}\right) \\ 0 & b_1-\beta\left(\frac{a_1-\rho}{a_2}\right)\end{array}\right]$
The characteristic equation is:
$\begin{aligned} \operatorname{Det}\left(J_3-\lambda I\right) & =\left|\begin{array}{cc}\alpha_1-a_1-\lambda & \rho\left(\frac{a_1-\alpha_1}{a_2}\right) \\ 0 & b_1-\beta\left(\frac{a_1-\alpha_1}{a_2}\right)-\lambda\end{array}\right|=0 \\ & =\rho_1-a_1-\lambda=0, \lambda_1=\rho_1-a_1\end{aligned}$
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 a1=5, a2=0.22, α=0.007, b1=3, b2=0.26, β=0.008 while the values of ρ1=3.5, β1=1.4 are randomly selected. N1and N2 are obtained based on Eqs. (7)-(8), λ1 and λ2 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 |
$-\Delta \rho_1$ |
N1, N2 |
2 |
$+\Delta \rho_1 \mathrm{n}$ |
N1, N2 |
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 |
N1, N2 |
|
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 inter-competition of the 2nd 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, N1 and dog size N2 is sought.
(ii) the impact of increase in pesticide, $+\Delta \rho_1$ , on ticks size of ticks, N1 and dog size N2 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 inter-competition of the 2nd species (β), on the population of competing species, N1 and N2.
(ix) the effect of the inter-competition 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, ρ1, on the populations of competing species, N1 and N2
$+\Delta \rho_1$ |
N1 |
N2 |
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, ρ1, on the populations of competing species, N1 and N2
$+\Delta \rho_1$ |
N1 |
N2 |
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, ρ1, 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 N1 decreases, the dynamical system is stable to a point, util it gets to a point when it becomes progressively unstable as N1 further approach zero.
Table 7 shows that as N1 decreases, the dynamical system is stable to a point, until it gets to a point when it becomes progressively unstable as N1 further approach zero.
In Table 8, a simultaneous decrease in the size of interacting species N1 and N2, the dynamical system is stable to a point, until it gets to a point when it becomes progressively unstable as N1 further approach zero.
Table 5. Impact of increasing the effects of pesticides, ρ1, 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, N2, 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, N1 and N2, 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 N2.
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 inter-competition of the 2nd species (β), on the population of competing species, N1 and N2
β |
N1 |
N2 |
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 inter-competition of the 2nd 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 first-order 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 second-order 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 tick-dog 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. Real-world 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.
[1] Gourbière, S, Morand, S, Waxman, D. (2015) Fundamental factors determining the nature of parasite aggregation in hosts. PLoS ONE, 10(2): e0116893. https://doi.org/10.1371/journal.pone.0116893
[2] Opanuga, A.A., Agboola, O.O., Okagbue, H.I., Oghonyon, J.G. (2015). Solution of differential equations by three semi-analytical techniques. International Journal of Applied Engineering Research, 10(18): 39168-39174.
[3] Agbolade, O.A., Anake, T.A. (2017). Solutions of first-order Volterra type linear integrodifferential equations by collocation method. Journal of Applied Mathematics, 2017: Article ID: 1510267. https://doi.org/10.1155/2017/1510267
[4] Moghaddar, S., Shorigeh, J., Gastrodashty, A.R. (2001). Prevalence of ectoparasites and its seasonal prevalence in dogs in Shiraz (Iran). In XII National Congress of Veterinary Parasitology, 62(S-2).
[5] Gonzalez-Miguel, J. (2022). Host-parasite relationship in veterinary parasitology: Get to know your enemy before fighting it. Animals, 12(4): 448. https://doi.org/10.3390%2Fani12040448
[6] Wilson, E.O. (2014). The Meaning of Human Existence. WW Norton & Company. ISBN 978-0.87140 -480-0
[7] Hirsch, M.W., Smale, S., Devaney, R.L. (2012). Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press.
[8] Ricardo, H.J. (2020). A Modern Introduction to Differential Equations. Academic Press.
[9] Karmaker, S., Ruhi, F.Y., Mallick, U.K. (2018). Mathematical analysis of a model on guava for biological pest control. Mathematical Modelling of Engineering Problems, 5(4): 427-440. https://doi.org/10.18280/mmep.050420
[10] Omudu, E.A., Atu, B.O., Ayashar, J. (2007). Epidemiological survey of canine babesiosis in Makurdi, Nigeria. Animal Research International, 4(3): 745-749. https://doi.org/10.4314/ari.v4i3.48685
[11] Walker, A.R., Koney, E.B.M. (1999). Distribution of ticks (Acari: Ixodida) infesting domestic ruminants in Ghana. Bulletin of Entomological Research, 89(5): 473-479. https://doi.org/10.1017/S0007485399000619
[12] Anderson, J.F. (2002). The natural history of ticks. Medical Clinics, 86(2): 205-218. http://doi.org/10.1016/S0025-7125(03)00083-X
[13] Magnarelli, L.A. (2009). Global importance of ticks and associated infectious disease agents. Clinical Microbiology Newsletter, 31(5): 33-37. http://doi.org/10.1016/j.clinmicnews.2009.02.001
[14] CDC (2020). How ticks spread disease. Centers for Disease Control and Prevention. https://www.cdc.gov/ticks/life_cycle_and_hosts.html
[15] Ewing, S.M. (2011). Bulldogs for dummies. Wiley Publishing. ISBN 978-0-7645-9979-8
[16] Freedman. A.H., Wayne, R.K. (2017). Deciphering the origin of dogs: From fossils to genomes. Annual Review of Animal Biosciences, 5: 281-307. https://doi.org/10.1146/annurev-animal-022114-110937
[17] Axelsson, E., Ratnakumar, A., Arendt, M.L., Maqbool, K., Webster, M.T., Perloski, M., Liberg, O., Arnemo, J.M., Hedhammar, A., Lindblad-Toh, K. (2013). The genomic signature of dog domestication reveals adaptation to a starch-rich diet. Nature, 495(7441): 360-364. https://doi.org/10.1038/nature11837
[18] Zanghi, B.M, Kerr, W., de Rivera, C., Araujo, J.A., Milgram, N.W. (2012). Effect of age and feeding schedule on diurnal rest/activity rhythms in dogs. Journal of Veterinary Behavior, 7(6): 339-347, https://doi.org/10.1016/j.jveb.2012.01.004
[19] Xhaxhiu, D., Kusi, I., Rapti, D., Visser, M., Knaus, M., Lindner, T., Rehbein, S. (2009). Ectoparasites of dogs and cats in Albania. Parasitology Research, 105: 1577-1587. https://doi.org/10.1007/s00436-009-1591-x
[20] Sahibi, H., Rhalem, A. (2007). Tiques et maladies transmises par les tiques chez les bovins au Maroc. Bull. Mens. Inf. Liaison PNTTA, 1-4.
[21] Estrada-Peña, A., Venzal, J.M., Kocan, K.M., Sonenshine, D.E. (2008). Overview: Ticks as vectors of pathogens that cause disease in humans and animals. Frontiers in Bioscience, 13: 6938-6946. https://doi.org/10.2741/3200
[22] Agbolade, O.M., Soetan, E.O., Awesu, A., Ojo, J.A., Somoye, O.J., Raufu, S.T. (2008). Ectoparasites of domestic dogs in some Ijebu communities, Southwest Nigeria. World Applied Sciences Journal, 3(6): 916-920.
[23] Dantas-Torres, F., Melo, M.F., Figueredo, L.A., Brandão-Filho, S.P. (2009). Ectoparasite infestation on rural dogs in the municipality of São Vicente Férrer, Pernambuco, Northeastern Brazil. Revista Brasileira de Parasitologia Veterinaria, 18(3): 75-77. https://doi.org/10.4322/rbpv.01803014
[24] Dantas-Torres, F. (2008). The brown dog tick, Rhipicephalus sanguineus (Latreille, 1806) (Acari: Ixodidae): From taxonomy to control. Veterinary parasitology, 152(3-4): 173-185. https://doi.org/10.1016/j.vetpar.2007.12.030
[25] Otranto, D., Dantas-Torres, F., Breitschwerdt, E.B. (2009). Managing canine vector-borne diseases of zoonotic concern: Part one. Trends in Parasitology, 25(4): 157-163. https://doi.org/10.1016/j.pt.2009.01.003
[26] Estrada-Peña, A., Ayllón, N., De La Fuente, J. (2012). Impact of climate trends on tick-borne pathogen transmission. Frontiers in Physiology, 3: 64. https://doi.org/10.3389/fphys.2012.00064
[27] Opara, M.N., Ezeh, N.O. (2011). Ixodid ticks of cattle in Borno and Yours truly, Obe states of Northeastern Nigeria: Breed and coat colour preference. Animal Research International, 8(1): 1359-1365.
[28] Hudson, P.J., Rizzoli, A.P., Grenfell, B.T., Heesterbeek, J.A.P., Dobson, A.P. (2002). Ecology of wildlife diseases. Oxford University Press, Oxford 1-5.
[29] Moriello, K.A. (2003). Zoonotic skin diseases of dogs and cats. Animal Health Research Reviews, 4(2): 157-168. http://doi.org/10.1079/AHRR200355
[30] Smith, F.D., Ballantyne, R., Morgan, E.R., Wall, R. (2011). Prevalence, distribution and risk associated with tick infestation of dogs in Great Britain. Medical and Veterinary Entomology, 25(4): 377-384. https://doi.org/10.1111/j.1365-2915.2011.00954.x
[31] Sahu, A., Mohanty, B., Panda, M.R., Sardar, K.K., Dehuri, M. (2013). Prevalence of tick infestation in dogs in and around Bhubaneswar. Veterinary World, 6(12): 982-985. https://doi.org/10.14202/vetworld.2013.982-985
[32] Zeleke, M., Bekele, T. (2004). Species of ticks on camels and their seasonal population dynamics in Eastern Ethiopia. Tropical Animal Health and Production, 36: 225-231. https://doi.org/10.1023/B:TROP.0000016830.30194.2a
[33] Krčmar, S., Ferizbegović, J., Lonić, E., Kamberović, J. (2014). Hard tick infestation of dogs in the Tuzla area (Bosnia and Herzegovina). Veterinarski arhiv, 84(2): 177-182.
[34] Prates, L., Otomura, F.H., Mota, L.T., Jean, M. (2013). Impact of antiparasitic treatment on the prevalence of ectoparasites in dogs from an indigenous territory, state of Parana, Brazil. Rev Patol Trop, 42(3): 339-351. https://doi.org/10.5216/rpt.v42i3.26923
[35] Opanuga, A.A., Edeki, S.O., Okagbue, H.I., Akinlabi, G.O., Osheku, A.S., Ajayi, B. (2014). On numerical solutions of systems of ordinary differential equations by numerical-analytical method. Applied Mathematical Sciences, 8(164): 8199-8207. http://dx.doi.org/10.12988/ams.2014.410807
[36] Edeki, S.O., Akinlabi, G.O., Adeosun, S.A. (2016). On a modified transformation method for exact and approximate solutions of linear Schrödinger equations. In AIP Conference Proceedings, 1705(1): 020048. http://dx.doi.org/10.1063/1.4940296
[37] Salih, D.A., El Hussein, A.M., Singla, L.D. (2015). Diagnostic approaches for tick-borne haemoparasitic diseases in livestock. Journal of Veterinary Medicine and Animal Health, 7(2): 45-56. https://doi.org/10.5897/JVMAH2014. 0345
[38] Chomel, B. (2011). Tick-borne infections in dogs—an emerging infectious threat. Veterinary Parasitology, 179(4): 294-301. https://doiorg/10.1016/J.Vetpar.2011.03.040
[39] Sumbria, D., Singla, L.D. (2017). Thwack of worldwide weather transformation on vector and vector-borne parasitic infections. ARC Journal of Animal and Veterinary Sciences, 3(2): 1-10.
[40] Liyanarachchi, D., Rajakaruna, R., Dikkumbura, A., Rajapakse, R. (2017). Ticks infesting wild, domestics animals and human of Sri-Lanka with new host records. Acta Tropica, 142: 291-204. https://doi.org/10.1016/j.actatropica.2014.11.001
[41] Taylor, M.A., Coop, R.L., Wall, R.L. (2015). Veterinary Parasitology. John Wiley & Sons.
[42] Otranto, D., Wall, R. (2008). New strategies for the control of arthropod vectors of disease in dogs and cats. Medical and Veterinary Entomology, 22(4): 291-302. https://doi.org/10.1111/j.1365-2915.2008.00741.x
[43] Dohoo, I.R., McDonell, W.N., Rhodes, C.S., Elazhary, Y.L. (1998). Veterinary research and human health. The Canadian Veterinary Journal, 39(9): 548-556.
[44] Adesina, O.S., Famurewa, O.K., Dare, R.J., Agboola, O.O., Odetunmibi, O.A. (2018). The mackey-glass type delay differential equation with uniformly generated constants. International Journal of Mechanical Engineering and Technology, 9(9): 467-477.
[45] Schnittger, L., Rodriguez, A.E., Florin-Christensen, M., Morrison, D.A. (2012). Babesia: A world emerging. Infection, Genetics and Evolution, 12(8): 1788-1809. https://doi.org/10.1016/j.meegid.2012.07.004
[46] James-Rugu, N.N., Iwuala, M.O.E. (2001). A study of the haemoparasites of dogs, pigs and cattle in Plateau State. Nigerian journal of science and Technology, 7: 20-27.
[47] Parola, P., Raoult, D. (2001). Ticks and tickborne bacterial diseases in humans: An emerging infectious threat. Clinical Infectious Diseases, 32(6): 897-928. https://doi.org/10.1086/319347
[48] Gray, J.S., Dautel, H., Estrada-Peña, A., Kahl, O., Lindgren, E. (2009). Effects of climate change on ticks and tick-borne diseases in Europe. Interdisciplinary Perspectives on Infectious Diseases, 2009: Article ID 593232. https://doi.org/10.1155/2009/593232
[49] Dantas-Torres, F., Chomel, B.B., Otranto, D. (2012). Ticks and tick-borne diseases: A one health perspective. Trends in Parasitology, 28(10): 437-446. https://doi.org/10.1016/j.pt.2012.07.003
[50] Homer, M.J., Aguilar-Delfin, I., Telford III, S.R., Krause, P.J., Persing, D.H. (2000). Babesiosis. Clinical Microbiology Reviews, 13(3): 451-469. https://doi.org/10.1128/cmr.13.3.451
[51] Agboola, O.O., Opanuga, A.A., Gbadeyan, J.A. (2015). Solution of third order ordinary differential equations using differential transform method. Global Journal of Pure and Applied Mathematics, 11(4): 2511-2517.
[52] George, I. (2019). Stability analysis of a mathematical model of two interacting plant species: A case of weed and tomatoes. International Journal of Pure and Applied Science, 9(1): 1-15.
[53] Galdi, G.P., Maremont, P. (2023). On the stability of steady-state solutions to the navier–stokes equations in the whole space. Journal of Mathematical Fluid Mechanics, 25: 7. https://doi.org/10.1007/s00021-022-00748-6
[54] Yan, Y., Ekaka-a, E.O.N. (2011). Stabilizing a mathematical model of population system. Journal of the Franklin Institute, 348(10): 2744-2758. https://doi.org/10.1016/j.jfranklin.2011.08.014
[55] Goeritno, A. (2021). Ordinary differential equations models for observing the phenomena of temperature changes on a single rectangular plate fin. Mathematical Modelling of Engineering Problems, 8(1): 89-94. https://doi.org/10.18280/mmep.080111