A Lotka-Volterra Non-linear Differential Equation Model for Evaluating Tick Parasitism in Canine Populations

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

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 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, 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.

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.