The purpose of this paper is to study the $\lambda$-dependence of the solution of the Lotka-Volterra problem:

There have been many studies on this subject. Differentiability with initial data for the functional differential equations was first established by Hale in [2], but differentiability with respect to delays for delay differential equations was proved by Hale and Ladeira in [3] and by A. Tămăşan in [13]. The paper of Hokkanen and Moroşanu [5] gives a proof for delay differential equations case using the step method. The Picard operators’ technique proposed by I.A. Rus [9], [10], [11], was used by V. Mureşanu [6] to prove continuity with respect to $\lambda$, M. Şerban [12], D. Otrocol [8] using the theorem of fibre contraction.

In this paper we use the following theorem for the simple case of ordinary differential system.

We consider the following Lotka-Volterra system with parameter:

We suppose that:

1. (H₁)

   $\;\tau_{1}\leq\tau_{2};\;t_{0}<b;\;J\subset\mathbb{R}$, a compact interval ;

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2. (H₂)

   $\;f_{i}\in C^{1}([t_{0},b]\times\mathbb{R}^{4}\times J),\;i=1,2$;

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3. (H₃)

   $\;\exists M>0$ such as $\big\|\frac{\partial f_{i}}{\partial u_{j}}(t,u_{1},u_{2},u_{3},u_{4};\lambda)\big\|_{\mathbb{R}^{n}}\leq M_{1},\;\newline$for all$\ t\in[t_{0},b],\;u_{j}\in R^{n},\;j=\overline{1,4},\;\lambda\in J,\ i=1,2$;

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4. (H₄)

   $\;\varphi\in C([t_{0}-\tau_{1},t_{0}]),\;\psi\in C([t_{0}-\tau_{2},t_{0}])$.

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In the above conditions, from Theorem 1 in [7] we have that the problem (5)–(8) has a unique solution, $(x_{1}(t;\lambda),x_{2}(t;\lambda))$.

We prove that

applying the Theorem 1 and using the step method.

We consider the system

with initial conditions

where $x_{1}\in C([t_{0}-\tau_{1},b]\times J)\cap C^{1}[t_{0},b],\;x_{2}\in C([t_{0}-\tau_{2},b]\times J)\cap C^{1}[t_{0},b]$.

We prove that

applying the Theorem 1 and using the step method.

For $t\in[t_{0},t_{0}+\tau_{1}]$, from the Theorem 1 we have that $x_{1}(t;\cdot)\in C^{1}(J),$ $x_{2}(t;\cdot)\in C^{1}(J)$.

For $t\in[t_{0}+\tau_{1},t_{0}+2\tau_{1}]$, from the Theorem 1 we have that $x_{1}(t;\cdot)\in C^{1}(J),\ x_{2}(t;\cdot)\in C^{1}(J)$. From (9) and $(H_{2})$ it follows that $x_{1}^{\prime}(t_{0}+\tau_{1}-0)=x_{1}^{\prime}(t_{0}+\tau_{1}+0)$ and $x_{2}^{\prime}(t_{0}+\tau_{1}-0)=x_{2}^{\prime}(t_{0}+\tau_{1}+0)$. Then $(x_{1}(t;\lambda),x_{2}(t;\lambda))$ is differentiable in $t_{0}+\tau_{1}$.

For $t\in[t_{0}+n\tau_{1},t_{0}+\tau_{2}]$, from the Theorem 1 we have that $x_{1}(t;\cdot)\in C^{1}(J),\ x_{2}(t;\cdot)\in C^{1}(J)$. From (9) and $(H_{2})$ it follows that $x_{1}^{\prime}(t_{0}+n\tau_{1}-0)=x_{1}^{\prime}(t_{0}+n\tau_{1}+0)$ and $x_{2}^{\prime}(t_{0}+n\tau_{1}-0)=x_{2}^{\prime}(t_{0}+n\tau_{1}+0)$. Then $(x_{1}(t;\lambda),x_{2}(t;\lambda))$ is differentiable in $t_{0}+n\tau_{1}$.

So $(x_{1}(t;\lambda),x_{2}(t;\lambda))$ is $C^{1}$ in the knots. ∎

We present below a simple example to illustrate the procedures of applying our results.

Equation (11) is the form of (9). Therefore, the assumptions $(H_{1})-(H_{4})$ are satisfied for (11)–(12). Moreover, the Theorem 2 hold. Following the proof of Theorem 3 we see that:

For $t\in[t_{0},t_{0}+1]$, from the Theorem 1 we have that $x_{1}(t;\cdot)\in C^{1}(J),\ x_{2}(t;\cdot)\in C^{1}(J)$.

For $t\in[t_{0}+1,t_{0}+2]$, from the Theorem 1 we have that $x_{1}(t;\cdot)\in C^{1}(J),\ x_{2}(t;\cdot)\in C^{1}(J)$. From (9) and $(H_{2})$ it follows that $x_{1}^{\prime}(t_{0}+1-0)=x_{1}^{\prime}(t_{0}+1+0)$ and $x_{2}^{\prime}(t_{0}+1-0)=x_{2}^{\prime}(t_{0}+1+0)$. Then $(x_{1}(t;\lambda),x_{2}(t;\lambda))$ is differentiable in $t_{0}+1$. So $(x_{1}(t;\lambda),x_{2}(t;\lambda))$ is $C^{1}$ in the knots. Therefore the Theorem 3 is proved.

Here we give a portrait of the trajectory of (11)–(12) drawn by a computer using MATLAB facilities. The results from numerical computation are plotted for $\lambda=1.35,1.37,1.39,1.41$ in Fig. 1.